more information – www.cambridge.org/9781107039902
LATTICE SUMS THEN AND NOW
The study of lattice sums began when early investigators wanted to go from
mechanical properties of crystals to the properties of the atoms and ions from
which they were built (the literature of Madelung’s constant). A parallel
literature was built around the optical properties of regular lattices of atoms
(initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many
famous scientists and mathematicians have delved into the properties of lattices,
sometimes unwittingly duplicating the work of their predecessors.
Here, at last, is a comprehensive overview of the substantial body of
knowledge that exists on lattice sums and their applications. The authors also
provide commentaries on open questions and explain modern techniques that
simplify the task of finding new results in this fascinating and ongoing field.
Lattice sums in one, two, three, four and higher dimensions are covered.
Encyclopedia of Mathematics and Its Applications
This series is devoted to significant topics or themes that have wide application
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the abstract theory is less important than a thorough and concrete exploration of
the implications and applications.
Books in the Encyclopedia of Mathematics and Its Applications cover their
subjects comprehensively. Less important results may be summarized as
exercises at the ends of chapters. For technicalities, readers can be referred to the
bibliography, which is expected to be comprehensive. As a result, volumes are
encyclopedic references or manageable guides to major subjects.
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A. Kushner, V. Lychagin and V. Rubtsov Contact Geometry and Nonlinear Differential Equations
L. W. Beineke and R. J. Wilson (eds.) with P. J. Cameron Topics in Algebraic Graph Theory
O. J. Staffans Well-Posed Linear Systems
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A. A. Borovkov and K. A. Borovkov Asymptotic Analysis of Random Walks
M. Deza and M. Dutour Sikiric Geometry of Chemical Graphs
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H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity
F. W. King Hilbert Transforms I
F. W. King Hilbert Transforms II
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H. Morimoto Stochastic Control and Mathematical Modeling
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and Engineering
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A. Kristály, V. D. Radulescu and C. Varga Variational Principles in Mathematical Physics, Geometry,
and Economics
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N. Vakil Real Analysis through Modern Infinitesimals
R. B. Paris Hadamard Expansions and Hyperasymptotic Evaluation
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L. Berlyand, A. G. Kolpakov and A. Novikov Introduction to the Network Approximation Method for
Materials Modeling
M. Baake and U. Grimm Aperiodic Order I
J. Borwein et al. Lattice Sums Then and Now
R. Schneider Convex Bodies: The Brunn–Minkowski Theory (2nd Edition)
E NCYCLOPEDIA OF M ATHEMATICS AND ITS A PPLICATIONS
Lattice Sums Then and Now
J. M. BORWEIN
University of Newcastle, New South Wales
M. L. GLASSER
Clarkson University, New York
R. C. McPHEDRAN
University of Sydney
J. G. WAN
Singapore University of Technology and Design
I. J. ZUCKER
King’s College London
University Printing House, Cambridge CB2 8BS, United Kingdom
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Information on this title: www.cambridge.org/9781107039902
c J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan and I. J. Zucker 2013
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2013
Printed in Great Britain By TJ International Ltd. Padstow Cornwall.
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Borwein, Jonathan M.
Lattice sums then and now / J. M. Borwein, University of Newcastle, New South Wales,
M. L. Glasser, Clarkson University, R. C. Mcphedran, University of Sydney, J. G. Wan,
University of Newcastle, New South Wales, I. J. Zucker, King’s College London.
pages cm. – (Encyclopedia of mathematics and its applications ; 150)
Includes bibliographical references and index.
ISBN 978-1-107-03990-2 (hardback)
1. Lattice theory. 2. Number theory. I. Title.
QA171.5.B67 2013
511.3 3–dc23
2013002993
ISBN 978-1-107-03990-2 Hardback
Additional resources for this publication at www.carma.newcastle.edu.au / LatticeSums
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Knowledge of lattice sums has been built by many generations of
researchers, commencing with Appell, Rayleigh, and Born. Two of the
present authorship (MLG and IJZ) attempted the first comprehensive review
of the subject 30 years ago. This inspired two more (JMB and RCM) to enter
the field, and they have been joined by a member (JGW) of a new generation
of enthusiasts in completing this second and greatly expanded compendium.
All five authors are certain that lattice sums will continue to be a topic of
interest to coming generations of researchers, and that our successors will
surely add to and improve on the results described here.
Contents
Foreword by Helaman and Claire Ferguson
Preface
1 Lattice sums
1.1 Introduction
1.2 Historical survey
1.3 The theta-function method in the analysis of lattice sums
1.4 Number-theoretic approaches to lattice sums
1.5 Contour integral technique
1.6 Conclusion
1.7 Appendix: Complete elliptic integrals in terms
of gamma functions
1.8 Appendix: Watson integrals
1.9 Commentary: Watson integrals
1.10 Commentary: Nearest neighbour distance
and the lattice constant
1.11 Commentary: Spanning tree Green’s functions
1.12 Commentary: Gamma function values in terms
of elliptic integrals
1.13 Commentary: Integrals of elliptic integrals, and lattice sums
References
2 Convergence of lattice sums and Madelung’s constant
2.1 Introduction
2.2 Two dimensions
2.3 Three dimensions
2.4 Integral transformations and analyticity
2.5 Back to two dimensions
2.6 The hexagonal lattice
2.7 Concluding remarks
2.8 Commentary: Improved error estimates
page xi
xvii
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viii
Contents
2.9 Commentary: Restricted lattice sums
2.10 Commentary: Other representations for
the Madelung’s constant
2.11 Commentary: Madelung sums, crystal symmetry, and Debye
shielding
2.12 Commentary: Richard Crandall and the Madelung constant for
salt
References
112
116
117
117
123
3 Angular lattice sums
3.1 Optical properties of coloured glass and lattice sums
3.2 Lattice sums and elliptic functions
3.3 A phase-modulated lattice sum
3.4 Double sums involving Bessel functions
3.5 Distributive lattice sums
3.6 Application of the basic distributive lattice sum
3.7 Cardinal points of angular lattice sums
3.8 Zeros of angular lattice sums
3.9 Commentary: Computational issues of angular lattice sums
3.10 Commentary: Angular lattice sums and the Riemann hypothesis
References
125
125
128
131
134
140
142
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154
4 Use of Dirichlet series with complex characters
4.1 Introduction
4.2 Properties of L-series with real characters
4.3 Properties of L-series with complex characters
4.4 Expressions for displaced lattice sums in closed form for
j = 2−10
4.5 Exact solutions of lattice sums involving indefinite quadratic
forms
4.6 Commentary: Quadratic forms and closed forms
4.7 Commentary: More on numerical discovery
4.8 Commentary: A Gaussian integer zeta function
4.9 Commentary: Gaussian quadrature
References
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157
157
161
5 Lattice sums and Ramanujan’s modular equations
5.1 Commentary: The modular machine
5.2 Commentary: A cubic theta function identity
References
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200
6 Closed-form evaluations of three- and four-dimensional sums
6.1 Three-dimensional sums
6.2 Four-dimensional sums
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Contents
6.3 Commentary: A five-dimensional sum
6.4 Commentary: A functional equation for a three-dimensional
sum
6.5 Commentary: Two amusing lattice sum identities
References
ix
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224
7 Electron sums
7.1 Commentary: Wigner sums as limits
7.2 Commentary: Sums related to the Poisson equation
References
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245
8 Madelung sums in higher dimensions
8.1 Introduction
8.2 Preliminaries and notation
8.3 A convergence theorem for general regions
8.4 Specific regions
8.5 Some analytic continuations
8.6 Some specific sums
8.7 Direct analysis at s = 1
8.8 Proofs
8.9 Commentary: Alternating series test
8.10 Commentary: Hurwitz zeta function
References
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264
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292
9 Seventy years of the Watson integrals
9.1 Introduction
9.2 Solutions for W F (w f ), W F (α f , w f ), and W S (ws )
9.3 The Watson integrals between 1970 and 2000
9.4 The singly anisotropic simple cubic lattice
9.5 The Green’s function of the simple cubic lattice
9.6 Generalizations and recent manifestations of Watson integrals
9.7 Commentary: Watson integrals and localized vibrations
9.8 Commentary: Variations on W S
9.9 Commentary: Computer algebra
References
294
294
299
303
306
310
312
315
316
319
320
Appendix
A.1 Tables of modular equations
A.2 Character table for Dirichlet L-series
A.3 Values of K [N ] for all integer N from 1 to 100
324
325
330
331
Bibliography
Index
350
364
Foreword by Helaman and Claire Ferguson
The Borwein Award: ‘Salt’, the sculpture, created in 2004
As sculptor, and also the inventor of the PSLQ integer relations algorithm, I
described to the Canadian Mathematical Society the sculpture expressing the
Madelung constant μ as follows:
μ :=
n,m, p
(−1)n+m+ p
n 2 + m 2 + p2
.
This polished solid silicon bronze sculpture is inspired by the work of David Borwein, his
sons and colleagues, on the conditional series above for salt, Madelung’s constant. This
series can be summed to give uncountably many constants; one is Madelung’s constant for
sodium chloride.
This constant is a period of an elliptic curve, a real surface in four dimensions. There
are uncountably many ways to imagine that surface in three dimensions; one has negative
Gaussian curvature and is the tangible form of this sculpture.
I will now explain some of the creative processes which led to this sculpture.
Actually, the inscription on the sculpture reads ‘created in 2004’ but, in the
spirit of this book Lattice Sums Then and Now, the creation started much earlier.
There are a couple of questions. First: why would a sculptor create a sculpture
about NaCl, as in ‘please pass the “nakkle” ’, sodium chloride or salt, a lifeessential mineral? Second: why would a sculptor be interested in Madelung’s
constant, a conditionally convergent series, subject to special summability, giving the electrostatic potential of the interpenetrating lattices of sodium (Na+ ) and
chlorine (Cl− ) ions?
The equals sign in the above equation is misleading at this stage because
the right-hand side is not defined as it stands. In fact, the right-hand side is a
conditional series of infinitely many positive and negative terms which can be
xii
Foreword
rearranged to give any real number whatsoever (!); the commutative law holds for
finitely many summands but does not hold for infinitely many summands.
I will answer these two questions raised above in order. First answer: I was born
in the Humbolt Basin in the Rocky Mountains and spent the first five years of my
life there. This basin contains the Great Salt Lake and huge areas of evaporated
deposits of salt minerals. At age three I saw my natural mother killed by lightning
and my natural father drafted into the Pacific theatre of World War II. Between
ages three and five I was the ‘guest’ of a large extended family of aunts and uncles.
After age five I was adopted by a carpenter and stone mason who lived in upstate
New York. There I learned to work with my hands. I was a strange little grass
orphan. The aunts wanted to mother me but the uncles had the pragmatic upper
hand. How strange was I? One aunt, in particular, recalled that she came in the
kitchen and found me at the kitchen table intent on sorting grains of salt. I had at
that age some sort of microscopic vision; some of my own children told me they
went through a sort of microscopic vision stage and later lost it, as did I. Those
little cubettes of salt had a great fascination for me, a fascination not shared by
sensible uncles. It was only much later that I learned to call the stuff Na+ Cl−
and that there was an interpenetrating pair of ion lattices underlying their cubical
structure which I certainly could not see. Even so it was interesting to stack those
grains of salt, the pre-Lego natural material I had to play with.
Second answer: I was an undergraduate at Hamilton College. My high school
mathematics teacher Florence Deci, who appreciated my art as well as my
Figure 1 The David Borwein Distinguished Career Award of the Canadian Mathematical
Society, created in 2004, is a bronze sculpture based on Benson’s formula for the
Madelung constant. An exact copy is given to each award winner. Photograph by
permission of the sculptor.
Foreword
xiii
maths, had advised me to choose a liberal arts school so I could do both art
and science. As a maths undergraduate I was fascinated by summability and
conditionally convergent series. From my Hamilton chemistry professor Leland
‘Bud’ Cratty, I first heard about salt and its curious connection with a mathematical sum, Madelung’s constant. The non-commutativity of infinitely many
summands was observed by Riemann in relation to the conditionally convergent
series k≥1 (−1)k−1 /k, which is supposed to be a representation of log 2 =
0.69314718 . . .; this value obtains by adding the terms in increasing k order, as is
implicit in the convention of summation notation. Even worse in some respects,
for Madelung’s series adding terms in increasing cubes gives a different answer
than adding terms in increasing balls. So what is the true value, the value with
which physicists like Born, Madelung, and Benson and mathematicians like the
Borweins would be satisfied? Benson answered this most remarkably with
μ = 12π
2 π
m≥0 n≥0 cosh ( 2
1
,
(2m + 1)2 + (2n + 1)2 )
which is an absolutely convergent series with all positive terms very rapidly
decreasing, affording its evaluation to many decimal places:
μ = 1.74756459463318219063621203554439740348516143662474175815282535076504 . . . ,
enough decimals to satisfy this sculptor. Subsequently, Borwein and Crandall [2]
and others have learned more and give an almost closed form for μ.
When the Borwein family asked me to do a sculpture about summability to
celebrate the mathematics of David Borwein and his sons, particularly its application to Madelung’s constant, you can see that my art and science pump had
been primed long ago in the deserts of the Rocky Mountains and the forests
of the Finger Lakes of upstate New York. It is true that when the Borweins
approached me about doing this sculpture, I had been celebrating mathematics
with sculpture for decades. However, they approached me while I was in my
negative-Gaussian-curvature phase and was carving granite, not salt, and would
my geometric negative Gaussian curvature phase be inhospitable to the hard
analysis about conditional triple sums over three-dimensional lattices?
It happened that I had developed a series of sculptures which involved twodimensional lattice sums, specifically having to do with the planets and Kepler’s
third law, i.e., that the squares of the orbital periods of the planets are proportional to the cubes of their radii, when this law is viewed in terms of elliptic
complex curves or real tori in four and three real dimensions. For example, the
planet Jupiter takes about y = 11 earth–sun years to elliptically orbit the sun at
x = 5 earth–sun distances, and 112 = y 2 = x 3 − x + 1 = 53 − 5 + 1 is a perfectly respectable Z-rank-2 elliptic curve in the two complex dimensions of x and
xiv
Foreword
y, which corresponds to four real dimensions. To get the planet Jupiter’s elliptic
curve into three real dimensions where I could expect to do sculpture required
negative-Gaussian-curvature forms and lattice sums!
Some mathematical details behind my negative-Gaussian-curvature phase
appeared in ‘Sculpture inspired by work with Alfred Gray: Kepler elliptic curves
and minimal surface sculptures of the planets’ [3], reflecting a keynote address by
Helaman and Claire Ferguson for the Alfred Gray Memorial Congress on Homogeneous Spaces, Riemannian Geometry, Special Metrics, Symplectic Manifolds
and Topology, held in September 2000 in Bilbao, Spain. This work actually made
copious use of and reference to the Borwein brothers’ Pi and the AGM [1], an
important resource for this negative-curvature phase of my sculpture.
What could be more natural than the conditional sum of a three-dimensional lattice as a period of a two-dimensional lattice to create a Madelung triply punctured
torus immersed with negative Gaussian curvature in three-dimensional space?
The Borwein Award sculpture emerged after considerable computational and
sculptural work, which I will sketch next.
I had some number-theoretical issues, which I discussed in detail with Jon Borwein. These involved the exponent in the denominator of the lattice sum, s = 12
for the square root. As a function of the complex variable s, ought not the series
L NaCl (s) have an analytic continuation to the whole plane, Riemann hypothesis,
and even a functional equation? I thought it important to immerse the matter of
salt symbolized by Madelung’s constant as L NaCl ( 12 ) in this larger world. Did it
have an Euler product? The answers to these two questions are yes, no, yes, and
no and appear in the writings of Jon Borwein and others elsewhere.
After much computation of L NaCl (s) for various values of s, I settled on μ =
L NaCl ( 12 ) and an elliptic curve,
y 2 = 4x 3 − (32.6024622677216 . . .)x − (70.6022720835820 . . .)
where the decimals correspond to two-dimensional lattice sums for a lattice
involving μ, with discriminant −99932.555 . . . The complex variables x, y are
complex numbers in four real dimensions and the complex curve equation
amounts to two real equations, so that the complex curve is really a surface in
four dimensions. There is a dimension embargo (the Planck length is even harder
to see than salt lattices!) on sculpture. Sculpture physically resides in spatial three
dimensions, hence I enjoy the use of negative curvature to get the geometric surface in four dimensions into the spatial three dimensions where I have much
experience.
While my aesthetic choice is to carve stone, my award sculptures are in polished silicon bronze. Silicon bronze is an alloy of copper with silicon and a few
other things to improve flow and polishing. A typical recipe for silicon bronze
is the ‘molecule’ 9438Cu + 430Si + 126Mn + 4Fe + Zn + Pb. I think of the
430Si + 126Mn + 4Fe + Zn + Pb piece as being the ‘stone’ part. I wonder, is
Foreword
xv
there a Madelung constant for this molecule, there being many loose ions in this
polycrystalline soup?
There are many steps in the casting of silicon bronze, but even before getting
to those, I had much computation to do in placing the complex curve into three
dimensions as a triply punctured torus. In the course of a computation and developing the computer graphics there are many choices to be made. My decision
process is informed by my studio experience in the same way that looking at
two-dimensional underwater video material is not at all the same after learning
to scuba dive in a three-dimensional environment. This is not the place to discuss all these transitions; there are many. In Figure 2 some of them are shown:
computer graphics, wire frame, clay, plaster. There are truly messy in-between
parts, especially making of the mould, the wax positive image, the ceramic shells
to form a negative flask, and a hot dry throat embedded in sand in which to pour
molten bronze; there is the high drama of the pouring of the bronze, the violence
of smashing the ceramic flask to release the imprisoned bronze, then the hackingoff of air escape sprues, chasing away all evidence of what violence the bronze
has experienced, grinding and sanding the bronze smooth enough to reveal the
inevitable natural errors, which must be excavated and welded in kind to prepare
for polishing. While the intermediate result is a beautiful polished bronze, shown
in Figure 3, this is not the end.
I am carving into this silicon bronze the name of each recipient of this elegant
CMS–SMC David Borwein Award, the provenance of the sculpture, and also in
Figure 1 something about the sculpture relating to salt and summability. This is
what is shown for the first recipient.
Art is always a social event in the end. In the case of this Borwein Award,
the truly priceless part is the awarding of a silicon bronze to celebrate the distinguished careers of gifted people who have given substantial parts of their lives to
creating new mathematics and even new mathematicians, as has David Borwein.
So far these people have included:
2010: Nassif Ghoussoub
2008: Hermann Brunner
2006: Richard Kane
Figure 2 Stages in the design of ‘Salt’.
xvi
Foreword
Figure 3 The David Borwein Distinguished Career Award of the Canadian Mathematical
Society. Photograph by permission of the sculptor.
I am honoured that my mathematical sculpture is part of recognizing and
celebrating mathematical lives.
References
[1] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number
Theory and Computational Complexity. Wiley, New York, 1987.
[2] J. M. Borwein and R. E. Crandall. Closed forms: what they are and why we care. Not.
Amer. Math. Soc., 60(1):60–65, 2013.
[3] Helaman Ferguson and Claire Ferguson. Sculpture inspired by work with Alfred Gray:
Kepler elliptic curves and minimal surface sculptures of the planets. Contemp. Math.,
288:39–53, 2000.
Preface
. . . Born decided to investigate the simple ionic crystal – rock salt (sodium chloride) – using
a ring model. He asked Landé to collaborate with him in calculating the forces between the
lattice points that would determine the structure and stability of the crystal. Try as they
might, the mathematical expression that Born and Landé derived contained a summation
of terms that would not converge. Sitting across from Born and watching his frustration,
Madelung offered a solution. His interest in the problem stemmed from his own research in
Goettingen on lattice energies that, six years earlier, had been a catalyst for Born and von
Karman’s article on specific heat. The new mathematical method he provided for convergence allowed Born and Landé to calculate the electrostatic energy between neighboring
atoms (a value now known as the Madelung constant).1 Their result for lattice constants of
ionic solids made up of light metal halides (such as sodium and potassium chloride), and
the compressibility of these crystals agreed with experimental results.2
The study of lattice sums is an important topic in mathematics, physics, and other
areas of science. It is not a new field, dating back at least to the work of Appell in
1884, and has attracted contributions from some of the most eminent practitioners of science (Born and Landé [1], Rayleigh, Bethe, Hardy, . . . ). Despite this,
it has not been widely recognized as an area with its own important tradition,
results, and techniques. This has led to independent discoveries and rediscoveries of important formulae and methods and has impeded progress in some topics
owing to the lack of knowledge of key results.
In order to solve this problem, Larry Glasser and John Zucker published in
1981 a seminal paper, the first comprehensive review of what was then known
about the analytic aspects of lattice sums. This work was immensely valuable to
many researchers, including the other authors of the present monograph, but now
1 More exactly, this energy can be obtained from the Madelung constant.
2 From [5], pp. 79–80. Max Born was the maternal grandfather of the singer and actress Olivia
Newton-John. Actually, soon after this they discovered that they had forgotten to divide by 2 in
the compressibility analysis. This ultimately led to the abandonment of the Bohr–Sommerfeld
planar model of the atom.
xviii
Preface
is out of date and lacks the immediate electronic accessibility expected by today’s
researchers.
Hence, we have the genesis of the present project, the composition of this
monograph. It contains a slightly corrected version of the 1981 paper of Glasser
and Zucker as well as additions reflecting the progress of the subject since 1981.
The authors hope it is sufficiently comprehensive in flavour to be of value to
both experienced practitioners and those new to the field. However, as the study
of lattice sums has applications in many diverse areas, the authors are well
aware that important contributions may have been overlooked. They would thus
welcome comments from readers regarding such omissions and hope that internet technology can make this a living and growing project rather than a static
compendium.
The emphasis of the results collected here is on analytic techniques for evaluating lattice sums and results obtained using them. We will nevertheless touch upon
numerical methods for evaluating sums and how these may be used in the spirit of
experimental mathematics to discover new formulae for sums. Those interested
primarily in numerical evaluation would do well to consult the relatively recent
reviews of Moroz [7] and Linton [6].
Several chapters in this monograph are based on published material and, as
such, we have tried to retain their original styles. In particular, we have not
attempted to iron out the differences in notation (we considered this option but
decided it would be very likely to introduce more errors and difficulties than it
removed). In particular, we alert the reader that several conventions of the sum
mation notation are used liberally throughout the monograph: the symbol
may
indicate either a single or a multiple sum and the variable(s) and range(s) of
summation may be omitted when they are clear from the context. The reader is
therefore advised to exercise caution when moving from chapter to chapter and to
note that various notations are listed at the beginning of the index. The index uses
bold font to indicate entries which are definitions and includes page numbers for
the various tables.
We have made a full-hearted attempt to correct misprints in the original material. The end-of-chapter commentaries also direct the reader to more recent
material and discussions of the source material. In the same spirit, each chapter has its own reference list while a complete bibliography is also provided at the
end of the book.
Chapter 1 originally appeared as [4]. Chapter 2 originally appeared as [3] and
is reprinted with permission from the American Institute of Physics. Chapter 8
originally appeared as [2]: it was published in the Transactions of the American
c the American Mathematical SociMathematical Society, in vol. 350 (1998), ety 1998. Chapter 9 originally appeared as [8]: it was published in the Journal
c Springer-Verlag 2011 with kind
of Statistical Physics, in vol. 134 (2011), permission from Springer Science+Business Media.
Preface
xix
Acknowledgments. The authors would like to acknowledge the following: the
late Nicolae Nicorovici, who created a TEX version of the original 1981 paper;
the Canada Research Chair Programme, the Natural Science and Engineering
Research Council of Canada, and the Australian Research Council, which have
supported the research of Jon Borwein and Ross McPhedran over many years;
also their many colleagues and students who have made important contributions –
David McKenzie, Graham Derrick, Graeme Milton, David Dawes, Lindsay Botten, Sai Kong, Chris Poulton, James Yardley, Alexander Movchan, Natasha
Movchan, and many others.
Chapter 1: The chapter authors wish to thank John Grindlay and Carl Benson for
aid with the literature search.
Chapter 2: The chapter authors would like to thank Professor O. Knop for originally bringing our attention to the problems involved in lattice sums. John Zucker
wishes to thank Sandeep Tyagi for his help and encouragement.
Chapter 9: The chapter author wishes to thank Geoff Joyce and Richard Delves,
since without their input none of this could have been written. He also thanks
Tony Guttmann for many pertinent comments.
Commentaries: We would like to thank Tony Guttmann, Roy Hughes, and
Mathew Rogers for their valuable feedback on these.
Website: The authors are maintaining a website for the book at www.carma.
newcastle.edu.au/LatticeSums/, at which updates and corrections can be received.
References
[1] M. Born and A. Landé. The absolute calculation of crystal properties with the help of
Bohr’s atomic model. Part 2. Sitzungsberichte der Königlich Preussischen Akademie
der Wissenschaften, pp. 1048–1068, 1918.
[2] D. Borwein, J. M. Borwein, and C. Pinner. Convergence of Madelung-like lattice
sums. Trans. Amer. Math. Soc., 350:3131–3167, 1998.
[3] D. Borwein, J. M. Borwein, and K. Taylor. Convergence of lattice sums and
Madelung’s constant. J. Math. Phys., 26:2999–3009, 1985.
[4] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and
Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980.
[5] N. T. Greenspan. The End of the Certain World: The Life and Science of Max Born.
Basic Books, 2005.
[6] C. M. Linton. Lattice sums for the Helmholtz equation. SIAM Review, 52:630–674,
2010.
[7] A. Moroz. On the computation of the free-space doubly-periodic Green’s function of the three-dimensional Helmholtz equation. J. Electromagn. Waves Appl.,
16:457–465, 2002.
[8] I. J. Zucker. 70 years of the Watson integrals. J. Stat. Phys., 143:591–612, 2011.
1
Lattice sums
Any attempt to employ mathematical methods in the study of chemical questions must be
considered profoundly irrational and contrary to the Spirit of Chemistry.
If Mathematical Analysis should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread
degeneration of that science.
Auguste Comte
Philosophie Positive (1830)
1.1 Introduction
It has been more than 100 years since Appell [2] introduced lattice sums into
physics, yet the article on which this chapter is based is apparently the first devoted
entirely to the subject. We are, of course, aware that parts of other reviews (such
as those by Born and Göppert-Mayer [21], Waddington [135], Tosi [133], and
Sherman [125]) have dealt with Coulomb sums in ionic crystals, as a casual reading of this chapter will demonstrate. In the perusal of 100 years of literature, we
will inevitably have missed or ignored relevant papers and their authors are urged
to communicate with us directly.
The organization of this review is as follows. In Section 1.2 we present a
historical survey, picking out and describing in detail some of the more important methods for calculation. Section 1.3 deals with the representation of lattice
sums as Mellin-transformed products of theta functions. In Section 1.4 we discuss
the evaluation of two-dimensional lattice sums by number-theoretic means, and
in Section 1.5 we examine a promising new application of contour integration.
Two brief appendices are concerned with connections among lattice sums, elliptic
integrals, and lattice Green’s functions.
2
Lattice sums
1.2 Historical survey
Lattice sums are expressions of the form
F(l)
(1.2.1)
l
where the vector l ranges over a d-dimensional lattice. In this review we shall be
concerned primarily with the sums for which
F(l) = (exp b · l)|l + g|−s .
(1.2.2)
We shall not indicate the value of d explicitly (although the cases d = 2, 3 will be
our main concern) nor shall we dwell on questions of convergence; these details
should be clear from the context.
Apparently, lattice sums were first considered in physics by Appell [2–4], who
examined the form of the solutions to Laplace’s equation for periodic sources. His
procedure was to Fourier-transform the potentials and solve the resulting algebraic
equations for the coefficients. He obtained an analytical expression for the potential of a line of point charges and derived a variant of the so-called Ewald method
for higher-dimensional arrays, but his work appears not to have reached the notice
of subsequent workers.
In 1902 Epstein [47] reported on efforts to find the most general function
satisfying a functional equation similar to that for the Riemann zeta function,
ζ (s) =
∞
1
.
ns
(1.2.3)
n=1
We describe what we feel are his most important findings.
Let q(l) be a positive definite quadratic form in the components of l:
q(l) = a11l12 + · · · + add ld2 + 2(a12l1l2 + · · · ),
where A = (ai j ), i, j = 1, . . . , d is a symmetric matrix. We shall denote by D
the determinant of A and by q(l) the quadratic form defined by the adjoint matrix
A = (a i j ),
ai j =
1 ∂D
.
D ∂ai j
Epstein found that the desired functions were1
g e−2πih·l
(q; s) =
Z ,
[q(l + g)]s/2
h l
(1.2.4)
(1.2.5)
where g and h are arbitrary vectors and the sum runs over the d-dimensional
integer lattice. The sum is assumed to omit l = −g if g is a lattice vector. This
1 The verticals indicate that the Epstein Zeta function Z depends on the vectors g and h as well as
on the scalars q and s.
1.2 Historical survey
3
class of Epstein zeta functions includes the solutions obtained by Appell and the
lattice sums which later became important in crystal physics.
Epstein showed that the functions (1.2.5) have the following properties:
g −g (i)
(q; s).
(q; s) = Z (1.2.6)
Z −h h g −g (ii)
(q; s).
(q; s) = Z (1.2.7)
Z h −h (iii) If any component of h is increased by an integer, the right-hand side of
(1.2.5) is unaffected.
(iv) If g j is increased by unity, (1.2.5) is multiplied by exp (−2πi h j ).
(v) The Epstein zeta functions may be continued analytically beyond their
domain of absolute convergence with respect to s (which is Re s > d) by
means of the integral representation
g ∞
−s/2
s/2−1
1
(q;
s)
=
( 2 s) Z dz
z
exp [−π zq(l + g) + 2πil · h]
π
1
h l
∞
dz z (d−s)/2−1
+ D −1/2 exp (−2πig · h)
1
×
exp [−π zq(k + h) − 2πig · k].
(1.2.8)
k
The sums which occur in (1.2.8) are the generalized theta functions
studied extensively by Krazer and Prym [96]. The two sums are related
by the analogue of the Jacobi transformation for ordinary theta functions
(see below). The lattice of the k vectors is, strictly speaking, the reciprocal
lattice for the l vectors, but in the present case these are identical. Equation
(1.2.8) is obtained by using the representation
∞
1
−s
z s−1 e−x z dz
(1.2.9)
x =
(s) 0
and breaking up the range of integration. This simple device, due to Riemann, with minor modifications is the underlying motif in most of the
practical work on lattice sums to be discussed below.
(vi) The Epstein zeta function obeys the functional equation
g h −2πig·h
d −s
(q; d−s).
(q; s) = e √
Z π −(d−s)/2 π −s/2 ( 12 s)Z 2
D
−g h (1.2.10)
4
Lattice sums
Equation (1.2.10) is valid for all sets of parameters, whereas (1.2.8) is
non-trivial only when g = 0, h = 0. However, if this is not the case,
(1.2.8) can be modified and becomes
0 ∞
2
2
−s/2
1
π
( 2 s)Z (q; s) =
dz z (s/2)−1
e−π zq(l)
√ − +
s
(s − d) D
1
0 l
∞
1
+√
dz z (d−s)/2−1
e−π zq(k) . (1.2.11)
D 1
k
(vii) When h is not an integer vector, (1.2.5) is an entire function of s; when h
is integral, (1.2.5) has a simple pole at s = d:
g 2π d/2
(q; s) = √
+ C0 + O(s − d).
Z (1.2.12)
D(d/2)(s − d)
h Epstein gave expressions for C0 , but these are quite complicated except
for the case d = 2, where the result was already known from earlier work
by Kronecker and Hurwitz (see Bellman [6]).
(viii) The zeta function vanishes for s = −2, −4, −6, . . . and for s = 0, unless
g is integral. In this case
g (q; 0) = − exp (−2πig · h).
(1.2.13)
Z h (ix) If the components of g and h are 0 or 1/2 and 4g · h is odd, then (1.2.5) is
identically zero for all s and q.
The numerical evaluation of lattice sums became an important topic in the first
two decades of the twentieth century, when X-ray studies, and particularly the
work of W. H. and W. L. Bragg, showed that ionic salts consist not of molecules
but of interpenetrating lattices of the corresponding ions. The formal aspects and
the role of lattice sums in crystal binding and lattice vibration studies were developed extensively by Born [17] and by his students, but the first accurate numerical
result pertaining to a three-dimensional crystal – rock salt – was obtained by
Madelung [103]. In view of its historical interest, we shall describe his calculation
in some detail.
Madelung considered the problem of calculating the potential at the origin of
an extended array of charges ±E arranged on a rock salt lattice, a portion of
which is displayed in Fig. 1.1. The lattice is considered to be composed of vertical
planes π0 (through the origin), π±1 , π±2 , . . ., and the plane π0 is considered to be
1.2 Historical survey
5
a
a
π2
a
π1
z
x
π0
L−1
y
O
L0
(a)
L1
+E
−E
(0, 1, 1)
(1, 1, 1)
(0, 0, 1)
(1, 0, 1)
a/2
(1, 1, 0)
(0, 1, 0)
(b)
(0, 0, 0)
(1, 0, 0)
Figure 1.1 (a) The rock salt structure decomposed into planes and lines of charges.
(b) Unit cell of the rock salt lattice.
formed from vertical lines L 0 (through the origin), L ±1 , . . . Hence, by symmetry,
the potential at O is
V (O) = VL 0 (O) + 2
∞
VL n (O) + Vπn (O) .
(1.2.14)
n=1
Thus the problem reduces to calculating the potential due to a line of alternating charges and a centred square planar array of alternating charges. (The prime
indicates that the charge E is absent from the origin on the line L 0 .) Although
unaware, apparently, of the earlier work by Appell, Madelung used precisely the
same method. Consider first the potential due to the array of charges shown in
Fig. 1.2a. An arbitrary linear periodic charge density can be written
ρ(x) =
∞
−∞
ρl e2πilx/a .
(1.2.15)
6
Lattice sums
a
E
E
−a
O
E
E
a
2a
x
r
(x, r)
(a)
a
−E
+E
−E
+E
−E
+E
−E
x
O
r
(x, r)
(b)
Figure 1.2 (a) Linear array of constant charges. (b) Alternating linear array.
Accordingly, we look for the potential at (x, r ) in the form
v1 (x, r ) =
∞
Fl (r )e2πilx/a .
(1.2.16)
−∞
By inserting (1.2.16) into Laplace’s equation, we find
d 2 Fl
4πl 2
1 d Fl
−
+
Fl (r ) = 0.
r dr
dr 2
a2
(1.2.17)
The solution to (1.2.17) which vanishes as r → ∞ (l = 0) is
Cl K 0 (2πlr/a), l = 0,
l = 0,
C0 ln (2a/r ),
Fl (r ) =
(1.2.18)
where K 0 denotes the modified Bessel function. However, from Poisson’s equation we have
1 ∂v1 ,
(1.2.19)
ρ(x) = −
2 ∂r r =0
which gives Cl = 2ρl . Hence the desired potential is
v1 (x, r ) = 2ρ0 ln
2a
r
+2
∞
−∞
ρl K 0
2πlr
a
e2πilx/a .
(1.2.20a)
(Note that we have adjusted the potential by an additive constant such that as
r → ∞ it gives the potential of a uniform line charge of density ρ0 .) For the
1.2 Historical survey
7
linear array in Fig. 1.2a Madelung’s result is
∞
E
2 + r 2 ]1/2
[(x
−
a)
−∞
∞
2πlr
2πlx
2E
2a
+2
cos
=
ρl K 0
ln
. (1.2.20b)
a
r
a
a
v1 (x, r ) ≡
l=1
Mathematically, this procedure is equivalent to using the Poisson summation formula, by which the slowly convergent first series in (1.2.20b) is replaced by
the rapidly convergent second series. The potential for the alternating array in
Fig. 1.2b is
v L (x, r ) = v1 (x, r ) − v1 (x + a/2, r )
∞
2π(2l − 1)r
2π(2l − 1)x
8E =
K0
cos
.
a
a
a
(1.2.21)
l=1
The potential at O due to all the other charges is most easily obtained by direct
summation:
1 1 1
4E
4E
1 − + − + ··· =
ln 2.
(1.2.22)
v L (O) =
a
2 3 4
a
In the case of the plane rectangular lattice shown in Fig. 1.3 we begin by looking
for a potential having the form
2πl2 x
2πl1 x
cos
(l = (l1 , l2 )).
V =
Fl cos
(1.2.23)
a
b
l
Substituting this into Laplace’s equation yields
d 2 Fl
− 4π 2 kl2 Fl = 0,
dz 2
kl2 =
l12
l22
+
,n
a2
b2
b
z
y
O
x
a
Figure 1.3 Planar rectangular grid of point charges.
(1.2.24)
8
Lattice sums
which has the solution (vanishing as z → ∞ for kl2 = 0)
C1 e−2π zkl ,
C0 + C1 z,
Fl =
kl = 0,
kl = 0.
This corresponds to the periodic charge density
2πl2 x
2πl1 x
ρ(x, y) =
cos
.
ρl cos
a
b
(1.2.25)
(1.2.26)
l
Proceeding from Poisson’s equation, just as for the linear case, we find
v2 (x, y, z) =
∞
4E e−2π kl z
2πl1 x
2πl2 y
cos
cos
ab
kl
a
b
l1 ,l2 =1
∞
2E 1 −2πlz/a
2πlx
2πly
−2πlz/b
+e
+
cos
cos
e
b
l
a
b
l1
2π E z
+ const.
−
ab
(1.2.27)
For a centred square alternating array, corresponding to the planes π in Fig. 1.1,
we obtain easily
Vπ =
∞ ∞
16E e−2π zkl
2πl1 x
2πl2 y
cos
cos
.
ab
kl
a
a
(1.2.28)
l1 =1 l2 =1
In addition to those above, Madelung’s paper contains a number of other
calculations relating to two-and three-dimensional periodic arrays.
Using formulas (1.2.21), (1.2.22), and (1.2.28), it is simple to calculate the
potential at the reference site O for the rock salt structure. On the basis of
(1.2.14), for three-decimal place accuracy Madelung found it sufficient to keep
the following contributions to v(O) = (E/a) μ.
The line L 0 , μ0 = 4 ln 2.
The lines L ±1 (x = 0, r = a/2), μ1 ∼
= 16K 0 (π ) + K 0 (3π );
The lines L ±2 (x = 0, r = a), μ2 ∼
= −16K 0 (2π ).
The lines L ±3 (x = 0, r = 3a/2), μ3 ∼
= 16K 0 (3π ).
The planes π±1 (x, y, z) = (0, 0, ±a/2),
√
√
exp (−π 2)
exp (−π 10)
∼
+2
μ4 = 32
.
√
√
10
2
√
exp (−π 2)
∼
.
(6) The planes π±2 (x, y, z) = (0, 0, ±a), μ5 = −32
√
2
(1)
(2)
(3)
(4)
(5)
Thus Madelung obtained μ ∼
= 3.487.
1.2 Historical survey
9
Although very expedient in this case, Madelung’s procedure is not generally applicable, owing to the complicated geometrical considerations needed to
decompose more complex lattice structures. Indeed, Landé [98] overlooked a
line of charges in decomposing the fluorite lattice. Madelung’s method was later
applied by Bethe [14], Jones and Dent [84], and others to obtain field distributions
in NaCl and other ionic crystals. It is interesting that, by summing directly over the
first nine shells of ions surrounding a sodium ion, Kendall [94] was able to obtain
Madelung’s result to within 10%. In this review we shall not consider such direct
summation procedures. Curiously, almost exactly when Madelung was doing his
work, Ornstein and Zernicke [112] independently used precisely the same method
in three dimensions to re-sum similar series occurring in a study of the magnetic
properties of cubic lattices.
The need for a more flexible summation procedure applicable to an arbitrary
crystal structure was met by P. Ewald [49]. The procedure is based on the rediscovery of the Riemann–Appell device expressed in Equation (1.2.11) and was
formulated and developed for cubic crystals in terms of ordinary theta functions
in Ewald’s 1912 thesis. He later published a generalized version, which, because
of its usefulness and wide application, we shall describe in detail. (For an exposition relating to d = 2, see Fetter [50].) Consider a crystal lattice described by the
vectors
R p = Rl + rt
p = (l, t),
t = 1, . . . , ν
(d = 3).
(1.2.29)
Here l is an integer vector which labels the Bravais lattice vectors Rl ; the vectors
rt lie in the unit cell and are associated with electric charges εt . We shall denote
by Kl the reciprocal lattice vectors corresponding to the Rl ; i.e., if
Rl = l1 a1 + l2 a2 + l3 a3
then
Kl = l1 b1 + l2 b2 + l3 b3 ,
where
ai · b j = δi j ,
V0 = a1 × a2 · a3 .
We shall illustrate the procedure for the electrostatic sum
ε p exp (i K 0 R pp ) exp (ik · R p ),
( p) =
R pp
(1.2.30)
p = p
where R pp = |R p − R p |.
We have
ν
( p) = lim
εt exp (ik · rt ) [π(r − rt ) − δtt Fl (r − rt )] , (1.2.31)
r→R p
t =1
10
Lattice sums
where
π(r) =
exp (i K 0 |Rl − r| + ik · Rl )
|Rl − r|
(1.2.32)
exp (i K 0 |Rl − r| + ik · Rl )
.
|Rl − r|
(1.2.33)
l
and
Fl (r) =
Hence, the problem reduces to calculating the slowly convergent sum (1.2.32) and
taking the limit in (1.2.31).
Precisely as Epstein [47] did, Ewald used the following basic theorem from
Krazer and Prym [96]:
Theorem 1.2.1 Let d1 , d2 , d3 generate a Bravais lattice, let A and v be arbitrary
vectors and let V0 = d1 × d2 · d3 . Let c1 , c2 , c3 be the corresponding unit vectors
for the reciprocal lattice and let
ql =
li di + v,
pl = π
li ci .
(1.2.34)
Then
exp (−ql2 + iql · A) =
l
π 3/2 exp [−(pl −
V0
1
2
v)2 + 2ipl · A],
(1.2.35)
l
where the sum is over the integer l lattice.
The sum in (1.2.32) is only conditionally convergent. To avoid technical difficulties attendant on this fact, it is advisable to render the series absolutely
convergent by including a convergence factor. This is done most simply by assuming that K 0 has an imaginary part that is small and positive. To transform (1.2.32)
we take note of the integral representation (1.2.10), for α ≥ 0,
∞
2
e−α R
2 2
2
2
=√
e−R t e−α /4t dt.
(1.2.36)
R
π 0
We can replace α by −i K 0 if the path of integration is deformed in such a way
that K 02 /4t 2 becomes purely imaginary as t approaches zero. Thus we have
∞
K 02
exp (i K 0 R)
2
2 2
exp −R t + 2 dt,
(1.2.37)
=√
R
4t
π (0)
where the path of integration leaves the origin along the ray arg t = arg K 0 − π/4
and then returns to the real axis. From (1.2.32) and (1.2.37) we have
∞
K2
2
exp −(Rl − r)2 t 2 + ik · Rl + 02 dt.
(1.2.38)
π(r) = √
4t
π (0)
l
1.2 Historical survey
11
Next, by (1.2.35) with ci = (t/π )ai , di = (π/t)bi , v = 2tr, so that
ql =
1
kl
(2π Kl + k) ≡ ;
2t
2t
we also have
2π
π(r) =
V0
∞
(0)
exp
l
−(k2l − K 02 )
dt
+ ikl · r 3 .
2
4t
t
(1.2.39)
The sum in (1.2.38) converges rapidly when t is large, while that in (1.2.39) converges best when t is small. Instead of dividing the range of the t-integration at 1,
as Epstein did, Ewald introduces the adjustable parameter η (assumed real) and
writes
π(r) = π (1) (r) + π (2) (r),
−(k2l − K 02 )
dt
2π η (1)
exp
+ ikl · r 3 ,
(1.2.40)
π (r) =
2
V0 (0)
4t
t
l
∞
K2
2
(2)
π (r) = √
exp −(Rl − r)2 t 2 + ik · Rl + 02 dt.
4t
π η
l
The path of integration in π (1) can be returned to the real axis and the integral is
elementary, while the integral in π (2) leads to the error function. In this way, we
obtain
4π exp [−(k2l − K 02 )/4η2 + ikl · r]
,
(1.2.41)
V0
k2l − K 02
l
exp (ik · Rl )
i K0
Re exp (i K 0 |Rl − r|) erfc |Rl − r|η +
.
π (2) (r) =
|Rl − r|
2η
π (1) (r) =
l
We also have, from (1.2.33) and the fact that erfc(x) = 1 − erf(x), for the term
t = t (rt = 0) in (1.2.31)
π(r) − Fl (r) = π (1) (r) + π (2) (r),
sin K 0 |Rl − r|
π (1) (r) = π (1) (r) −
|Rl − r|
1
i K0
Re exp (i K 0 |Rl − r|) erf |Rl − r|η +
,
−
|Rl − r|
2η
π (2) (r) = π (2) (r)
i K0
exp (ik · Rl )
Re exp (i K 0 |Rl − r|) erfc |Rl − r|η +
.
−
|Rl − r|
2η
(1.2.42)
12
Lattice sums
There is now no difficulty in taking the limit in (1.2.31). The quantity η is to be
chosen so that both series converge rapidly. For the case of a Madelung sum we
take both K 0 and k → 0, so
( p) → V (R p ) =
ν
t =1
π (1) =
2εt η
εt (π (1) + π (2) ) − √ ,
π
1 exp [−π 2 K l2 /η2 + 2πiKl · (rt − rt )]
,
π V0
K l2
(1.2.43)
l
π (2) =
erfc(|R p − R p |η)
.
|R p − R p |
p
The prime on the sum π (1) indicates that the l = (0, 0, 0) term is omitted because,
for an ionic crystal,
ν
εt = 0.
t =1
We have also used the fact that exp (2πiKl · R p ) = 1. Equations (1.2.43) are
very useful for practical calculations, as we shall see by evaluating the Madelung
constant for NaCl.
We have
V (O) = V (1) (O) + V (2) (O),
1 exp (−π 2 K l2 /η2 ) 2ηE
V (1) (O) =
Sl
+ √ ,
π V0
π
K l2
l=0
Sl =
2
εt exp (−2πiKl · rt ), n
t=1
V (2) (O) =
2 εt
t=1 l=0
erfc(|Rl + rt |η)
.
|Rl + rt |
For the NaCl structure (see Fig. 1.1)
a1 =
1
2
b1 =
1
a (1, 1, 1),
a(0, 1, 1),
a2 =
1
2
b2 =
1
a (1, 1, 1),
V0 =
1
4
a(1, 1, 0),
a3 =
1
2
b3 =
1
a (1, 1, 1).
a(1, 0, 1),
a3
Hence,
1
a(l2 + l3 , l1 + l2 , l1 + l3 ),
2
1
Kl = (−l1 + l2 + l3 , l1 + l2 − l3 , l1 − l2 + l3 ).
a
Rl =
(1.2.44)
1.2 Historical survey
13
The basis in the unit cell is
r1 = (0, 0, 0)
(+E),
r2 =
1
a(1, 1, 1)
2
(−E).
It is convenient to replace η by η/a. To evaluate V (1) (O), we note that
Sl =
2E, l1 + l2 + l3 odd,
0
otherwise.
Thus we sum only over those reciprocal lattice vectors for which l1 +l2 +l3 is odd.
For example, the six vectors l = [1, 0, 0] together with (1, 1, 1) and (−1, −1, −1)
give the eight reciprocal lattice vectors aKl = [1, 1, 1] for which a 2 K2l = 3. In
this way, we easily compile the following table:
a 2 K2l
Number
3
11
19
√
(η = 2/ π )
Summand in V (1) (O)
8
24
24
0.3218078
0.0004917
0.0000005
Hence V (1) (O) ∼
= (−E/a)(3.3554000). By inspection it can be seen that the term
t = 1 in V (1) (O) involves a sum over the vectors m = (l2 + l3 , l1 + l2 , l1 + l3 )
for which m 1 + m 2 + m 3 is even, while the term t = 2 correspondingly involves
the vector m = (l2 + l3 + 1, l1 + l2 + 1, l1 + l3 + 1) for which the sum of the
√
components is odd. Therefore we have (for η = 2/ π)
V
(2)
√
2E m 1 +m 2 +m 3 erfc( π m)
.
(O) =
(−1)
a
m
(1.2.45)
m=0
We easily construct the corresponding table:
m
[1, 0, 0]
[1, 1, 0]
[1, 1, 1]
[2, 0, 0]
Number
6
12
8
6
m
Sign
√1
√2
3
2
−
+
−
+
Summand in V (2) (O)
√
(η = 2/ π )
−0.1462648
+0.0066688
−0.0001320
+0.0000034
from which we have V (2) (O) = −(E/a)(0.13972471) and μ = 3.49512. The
√
value η = 2/ π has been chosen so both series converge at roughly the same rate.
Although it is not of direct concern, it is interesting to consider a paper by
Born [18], where the following problem is discussed: Arrange equal numbers
of positive and negative charges on a simple cubic lattice of finite extent so
14
Lattice sums
the electrostatic energy is a minimum. Born treats this problem subject to the
following simplifications:
(1) periodic boundary conditions are used;
(2) the Coulomb potential is replaced by e−κr /r with κ → 0 eventually;
(3) Ewald’s method is used for computation.
Again, we are dealing with an integer lattice, but the charges at the lattice sites
have period N in all directions:
εl+(N ,0,0) = εl+(0,N ,0) = εl+(0,0,N ) = εl ;
(1.2.46)
we have denoted the sites in the ‘cell’ by l0 , . . . , l N −1 . By electrical neutrality
N
−1
εl j = 0.
j=0
The electrostatic potential at l j is
j
=
εl+l j
l
l=0
e−κl ,
(1.2.47)
so the electrostatic energy per cell becomes
=
N −1
1
εl j
2
j
=
(1.2.48a)
l=0
j=0
sl =
1 e−κl
sl
,
2
l
N
−1
εl j εl+l j .
(1.2.48b)
j=0
Note that is equivalent to the potential at the origin of an infinite lattice with
‘charges’ sl at each site. These charges have the properties
sl+(N ,0,0) = sl+(0,N ,0) = sl+(0,0,N ) = sl = s−l .
Consequently they have the Fourier representation
sl =
N
−1
ξ j cos
j=0
2π
(l · l j ),
N
(1.2.49)
where
ξj =
N −1
1 2π
(l j · li ).
sli cos
3
N
N
i=0
(1.2.50)
1.2 Historical survey
15
Therefore (letting κ → 0)
=
N −1
1 ξj
N3
j,
(1.2.51)
j=0
where
j
=
1 3 cos (2π/N )(l · l j )
N
,
2
l
j = 0, 1, . . . , N − 1,
(1.2.52)
l=0
are called normal potentials. It was suggested by Born, and later made explicit by
Emersleben [40] and Hund [81], that any ionic crystal can be approximated by a
simple cubic lattice with point charges at various ‘rational’ sites in the unit cell.
Hence all Madelung constants can be evaluated simply as finite sums once one
knows the value of the function (Born’s Grundpotential)
1
cos 2π(l · z).
(1.2.53)
π(z) =
l
l=0
This has the periodicity properties
π(z 1 + 1, z 2 , z 3 ) = π(z 1 , z 2 + 1, z 3 ) = π(z 1 , z 2 , z 3 + 1),
(1.2.54)
π(z) = π(−z),
and so by symmetry need only be calculated in one-fortyeighth of the cubic unit
cell. On the basis of these considerations, Born showed that the NaCl structure
represents a relative minimum for the electrostatic energy. There is yet no general proof that it provides the absolute minimum. (Born did prove this for one
dimension.)
Born turned over the problem of evaluating the Grundpotential to his student
Emersleben, who devoted his thesis to this topic, and the results were summarized
in two papers in 1923 [40, 41]. In the first paper Emersleben extended the Grundpotential idea to an arbitrary Bravais lattice and to the case of a general inverse
power law; he showed these quantities to be simple Epstein zeta functions:
0 π(z) = Z (q; s),
z where the form q is given by the unit vectors of the lattice,
q = (ai j ) = (ai · a j ).
In addition, Emersleben [40] generalized Epstein’s integral representation (1.2.8)
slightly, by using Ewald’s idea of decomposing the range of the z-integral at z = η
rather than z = 1. It is interesting that this is the first publication in which Epstein’s
work is referred to in the context of ionic lattice sums, although apparently no
16
Lattice sums
use was made of his results other than to assess continuity with respect to s.
In his second paper, Emersleben [41] applied Ewald’s procedure to calculate
π(m 1 /12, m 2 /12, m 3 /12), 0 ≤ m i ≤ 6, and used the results to find the Madelung
constants for several cubic crystals. For example, for the NaCl structure shown in
Fig. 1.1b, we have eight ions in a unit cell with
+E:
l0
l1
l2
l3
= (0, 0, 0)
= (0, 1, 1)
= (1, 0, 1)
= (1, 1, 0)
−E:
l4
l5
l6
l7
= (1, 1, 1)
= (1, 0, 0)
= (0, 1, 0)
= (0, 0, 1)
ξ0 = ξ1 = ξ2 = ξ3 = ξ= ξ6 = ξ7 = 0,
ξ4 = 8E.
From (1.2.51) and (1.2.53),
1 ξj
π
=−
4
E
7
μNaCl
j=0
l j1 l j2 l j3
,
,
2 2 2
= −2π
1 1 1
, ,
2 2 2
= 3.495115.
In this way Emersleben [41] found the following values:
Crystal
Madelung constant μ
ZnS
7.56584 = π( 14 , 14 , 14 ) + 34 π(0, 0, 12 ) + 14 π( 12 , 12 , 12 )
CaF2
5.81828 = 12 π( 14 , 14 , 14 ) + 34 π(0, 0, 12 ) + 14 π( 12 , 12 , 12 )
CsCl
2.03536 = 32 π(0, 0, 12 ) + 12 π( 12 , 12 , 12 )
Cu2 O
3 π(0, 0, 1 ) + 3 π(0, 0, 1 ) + 1 π( 1 , 1 , 1 )
4.75219 = 32
4
2
2
4
2 2 2
3 π(0, 1 , 1 ) + 1 π( 1 , 1 , 1 )
+ 34 π(0, 14 , 14 ) + 16
2 4
8
4 4 4
3
1
1
1
3 π( 1 , 1 , 1 )
+ 4 π( 2 , 2 , 2 ) + 32
4 2 2
As pointed out by Hund [81] the result for Cu2 O is incorrect, but it has often
found its way into the modern literature. The correct result is given in Table 1.2
below.
At the same time another of Born’s students, H. Kornfeld [95], adapted Ewald’s
procedure to the calculation of the electrostatic energy of multipole lattices. The
Ewald procedure continues to be the most widely used numerical scheme for calculating lattice sums and particularly Madelung constants. A survey of numerical
values obtained in this way before 1932 was given by Sherman [125] (see also
Waddington [135] and Tosi [133]).
A third approach to calculating lattice sums was given by Jones [83] and
Jones and Ingham [85], who were concerned with the evaluation of Epstein zeta
1.2 Historical survey
17
functions for large integral s, especially for the cubic lattices. These arise naturally
in examing the stability of rare gas crystals whose atoms interact via the 6–12
potential, for example.2 The method is based on E. Landau’s results [97] relating to the distribution of lattice points within ellipsoids and particularly on the
formula
{N − q(l + q)}σ e2πih·l ν(N − q)
(1.2.55)
Q σ (N ) ≡
l
=
(σ + 1){h}
√ π d/2 N σ +d/2 − {g} N σ e2πig·h + Q σ (N ),
1
(σ + 1 + 2 d) D
where
Q σ (N ) = e−2πig·h
N (σ/2)+(d/4) e−2πig·k
Jσ +d/2 [2πq(k + h)N ]1/2
√
[q(k + h)]σ/2+d/4
πσ D
k
(1.2.56)
and Jν denotes the Bessel function of the first kind. Here ν(N − q) denotes that
the sum is restricted to vectors l for which q(l + g) is less than N ; {h} is 1 if h is
an integer vector and vanishes otherwise.
From the fact that
N
s 2πih·l
e2πih·l ν(N − q)(q −s/2 − N −s/2 ) =
e
ν(N − q)
u −s/2−1 du
2
q
l
l
N
s
u −s/2−1 Q 0 (u) du,
(1.2.57)
=
2 0
we immediately have the representation
q ∞
(q; s) = s
Z u −s/2−1 Q 0 (u) du
2 0
h and
e
2πih·l
ν(N − q) q
−s/2
l
=N
−s/2
s
Q 0 (N ) +
2
N
(1.2.58a)
u −s/2−1 Q 0 (u) du.
0
(1.2.58b)
By subtracting the last two relations we obtain
g ∞
(q; s) = Z N − N −s/2 Q 0 (N ) + s
u −s/2−1 Q 0 (u) du,
Z 2
0
h e2πih·l ν(N − q) q −s/2 .
(1.2.59)
ZN =
l
2 The 6–12 potential is given by V (r ) = A/r 6 + B/r 12 , where A and B are material constants.
12
12
12
18
Lattice sums
Next we integrate by parts p times, which gives
g
p n + s/2 − 1
Z (q; s) = Z N −
N−s/2−n Q n (N )
n
h
n=0
s/2 + p − 1 ∞
s
+p
u − p−s/2−1 Q p (u) du. (1.2.60)
+
2
p
0
Now using Landau’s representation (1.2.55) we obtain Jones’ and Ingham’s
working formula
g p + s/2 − 1
{h} p!π d/2 (s + 2 p)
N −(s−d)/2
Z
(q; s) = Z N + ( p + d/2 + 1)√ D(s − d)
p
h p + s/2 − 1
−2πig·h s + 2
N −s/2
− {g}e
s
p
p n + s/2 − 1
Q n (N )N −s/2−n + R p (N , s), (1.2.61)
−
n
n=0
where R p is due to the term Q p (N ) in (1.2.56). Finally, technical lemmas are
provided which in essence show that, for p > d/2, N > 12, p + d/2 − 1/2 < 6,
h s(s + 2) · · · (s + 2 p)
q; p + d + 3 .
Z |R p (N , s)| <
√
2
2
2 p π p+2 D N s/2+ p/2−d/4+3/4 0 (1.2.62)
The corrections to the finite sum Z N on the right-hand side of (1.2.61) are used
merely to estimate the value of N for which Z N gives the zeta function to the
required accuracy. For s ≥ 6, N = 25 was found to be sufficient for six-place
accuracy.
To evaluate Z N and Q σ (N ), number-theoretical tables were used for the
quantity
an (q) = number of representatives of n by the quadratic form q,
so that
ZN =
N
n −s/2 an (q),
n=1
(1.2.63)
N
Q σ (N ) =
(N − n)σ an (q).
n=1
In all the cases considered these series required that no more than five terms be
retained for the necessary accuracy.
1.2 Historical survey
19
As an example of the use of these formulae we shall work out the sum
R −12
S=
for a face-centred cubic lattice, in which case
S=
[(l1 + l2 )2 + (l2 + l3 )2 + (l1 + l3 )2 ]−6
l=0
=
26
a 12
0 Z (q; 12),
0 (1.2.64)
where q(p) = l12 +l22 +l32 +l1l2 +l2l3 +l3l1 . (Thus q(l) = 32 l12 + 32 l22l32 −l3l1 −l1l2 ,
D = 12 .) Using p = 3 (we estimate z| 00 |(q; 6) < 20) we easily find that |R3 | <
106 with N = 10. Hence, to calculate Z 10 and Q n (10), we require the first ten
values of an (q). But it is clear that q(l) = 12 (m 21 + m 22 + m 23 ), where m 1 = l1 + l2 ,
m 2 = l2 + l3 , m 3 = l3 + l1 so that the sum m l + m 2 + m 3 is even. Therefore an (q)
is the number of ways in which 2n can be expressed as the sum of three squares
(since m 1 + m 2 + m 3 is even only if m 21 + m 22 + m 23 is even, this restriction can
be dropped). By inspection we have
a1
a2
a3
a4
a5
= 12,
= 6,
= 24,
= 12,
= 12,
a6
a7
a8
a9
a10
= 8,
= 48,
= 6,
= 36,
= 48.
Thus from (1.2.63)
Z 10 = 12.13109,
Q 0 (10) = 164,
and therefore
Q l (l O) = 680,
Q 2 (10) = 3884,
Q 3 (10) = 26036
0 (q; 12) = 12.13137.
Z 0 This method is reasonable for simple lattice structures if a desk calculator is
used, but does not appear to be suitable for large-scale computation. Extensive
use of Jones’ and Ingham’s results was made by Hund [80] in his study of the stability of cubic lattices. Hund appears to have been the first to use the expedient of
estimating the remainder, after summing over the first few shells of nearest neighbours, by replacing the sum by an integral. This procedure was developed more
carefully by Evjen [48], but its accuracy is questionable. Finally, Topping [132]
modified Jones’ and Ingham’s procedure to evaluate several two-dimensional lattice sums. However, as will be discussed later in detail, all such sums can be
20
Lattice sums
evaluated exactly. Indeed, precisely the sums considered by Topping had been
exactly evaluated by Lorenz [100] more than 50 years earlier.
The lattice summation problem does not appear to have attracted much attention over the decade from 1930 to 1940. In spite of excellent reviews of the
subject by Born and Göppert-Mayer [21] and by Højendahl [77], and extensions
of the Ewald procedure such as that by Fuchs [51], who used it in calculating
the electrostatic energy in metallic copper, as late as 1939 Orr wrote: ‘Since no
simple analytic method is available, for evaluating these sums, they were summed
numerically over at least the nearest 250 points and the remaining contribution
determined by integration.’ The most significant work during this period was
Hund’s [81] extension of Born’s Grundpotential concept.
Hund’s basic idea was to decompose a given lattice structure into ν Bravais
lattices, each of which consists of identical ions. He then calculated the potential at a given site due to each of these simple lattices and summed over these
contributions to obtain the final result. However, since the component sums are
individually divergent, the corresponding lattice was assumed to be rendered electrically neutral by a uniform charge distribution of the appropriate sign. These
fictitious charge distributions naturally cancel and make no contribution to the
final result. This procedure has the advantage that, since for a given lattice system
the lattice constants can be scaled out of the individual sums, the lattice sums for
all crystals can be expressed in terms of a few basic dimensionless ones. Thus,
relations between the sums for the three basic cubic lattices, for example, can
be discerned immediately. Secondly, once these basic sums have been calculated,
e.g., by the Ewald method, the lattice sums for any cubic crystal can be evaluated. This is also the case, with minor modification, for other crystal systems.
For example, in the case of hexagonal crystals the basic sums depend on the c/a
ratio, which must be the same for crystals whose lattice sums are to be compared. Furthermore, symmetry can be exploited to reduce the number of sums to
be calculated even further.
To illustrate this, consider the Madelung sum
S =
t
ν
l
t=1
eZ t
,
|Rl + rt − rt |
(1.2.65)
where the prime indicates that the term corresponding to the vanishing of the
denominator is omitted and e denotes the electronic charge. This sum can be
expressed in the form
e
Z t ψ(0, 0, 0) +
St =
Z t ψ(xt , yt , z t ) ,
(1.2.66)
a
t=t
where, in terms of the unit vectors,
a1 + yt
a2 + z t
a3 ,
rt − rt = xt
1.2 Historical survey
21
is a point in the unit cell. Here ψ(x, y, z) is the potential at the point a −1 (x
a1 +
a3 ) of a Bravais lattice of unit charges at
y
a2 + z
1
a1 + l2
a2 + l3
a3 )
(l1
a
neutralized by a uniform negative charge distribution. The potential ψ(0, 0, 0) is
the same quantity but with the term l = (0, 0, 0) omitted. By the use of symmetry
it is easily shown that ψ(x, y, z) (see, e.g., Tosi [133]) need be calculated only in
some irreducible symmetry element of the unit cell. The quantity ψ(0, 0, 0) can
be eliminated for the cubic lattices as follows.
The simple cubic lattice of side a can be looked on as composed of eight simple
cubic lattices of side 2a. The coordinates of a point (x, y, z) in the unit cell of the
former has the respective coordinates in the unit cell of the latter
ξ1
ξ2
ξ3
ξ4
= ( 12 x, 12 y, 12 z),
= ( 12 (x − 1), 12 y, 12 z),
= ( 12 x, 12 (y − 1), 12 z),
= ( 12 x, 12 y, 12 (z − 1)),
ξ5
ξ6
ξ7
ξ8
= ( 12 (x − 1), 12 (y − 1), 12 z),
= ( 12 (x − 1), 12 y, 12 (z − 1)),
= ( 12 x, 12 (y − 1), 12 (z − 1)),
= ( 12 (x − 1), 12 (y − 1), 12 (z − 1)).
(1.2.67)
Therefore we have Hund’s identity
1
ψ(ξk ).
2
8
ψ(x, y, z) =
(1.2.68)
k=1
In particular,
ψ(0, 0, 0) = 12 [ψ(0, 0, 0) + ψ( 12 , 0, 0) + ψ(0, 12 , 0) + ψ(0, 0, 12 )
+ ψ( 12 , 12 , 0) + ψ( 12 , 0, 12 ) + ψ(0, 12 , 12 ) + ψ( 12 , 12 , 12 )]. (1.2.69)
Hence, by cubic symmetry,
ψ(0, 0, 0) = 3ψ( 12 , 0, 0) + 3ψ( 12 , 12 , 0) + ψ( 12 , 12 , 12 ).
This procedure has been significantly elaborated and extended and will be
discussed in more detail in Section 1.3.
In the early 1940s a number of lattice sums were calculated by Born and coworkers (Born and Fürth [20]; Born and Misra [22]), by Misra [106], and by Peng
and Powers [113], principally by the Ewald technique. One exception is the work
of Born and Bradburn [19], who calculated a variety of sums having the form
Snk (α) =
l k1 l k2 l k3
1 2 3
n
l=0
l
exp (−il · α)
(1.2.70)
for the face-centred cubic lattice. These were obtained by means of the identity
Snk (α) = i k1 +k2 +k3
∂ k1 +k2 +k3
∂αk11 ∂αk22 ∂αk33
Sn0 (α),
(1.2.71)
22
Lattice sums
so that only Sn0 (α) need be evaluated in detail. The latter was treated by a variant
of the Epstein–Ewald method, as we now illustrate for the simple cubic lattice.
By using the representation (1.2.9) we have
∞
1
2
Sn0 (α) =
e−il·α
e−l μ μn/2−1 dμ
(n/2)
0
l=0
∞
n/2
(π )
2
β n/2−1
e−πβl −il·α dβ.
(1.2.72)
=
(n/2) 0
l=0
In terms of Jacobian theta functions, the sum above is
3
i=1
θ3
α
i
2
, e−πβ − 1 = σ (β).
(1.2.73)
Next, we break up the integral into a part from 0 to 1 and a part from 1 to ∞. The
part from 1 to ∞ may be obtained directly from the integral in (1.2.72), but for
the part from 0 to 1 we first use Jacobi’s transformation
θ3 (α, e−πβ ) = β −1/2 exp (−α 2 /πβ) θ3 (iα/β, e−πβ ).
(1.2.74)
We now have
π n/2
2
S0 + S1 −
,
(n/2)
n
∞
S0 =
β n/2−1 σ (β) dβ,
Sn0 (α) =
1
1
S1 =
0
(1.2.75)
3
iα j −πβ
α2 .
,e
β n/2−5/2 exp −
θ3
4πβ
2β
j=1
The integral S0 is given by
S0 =
∗
πl 2
cos (l · α)
4
−n/2
n πl 2
,
2 4
,
(1.2.76)
where the symbol ∗ denotes that only one of l and −l is included and (a, x) is
the incomplete gamma function. Next we have, with β −1 → t,
2 ∞
t α
dt
+ lπ
t −n/2+1/2 exp −
S1 =
π 2
1
l
[(α/2) + lπ ]2 (n−1)/2 1 n (α/2 + lπ )2 =
. (1.2.77)
− ,
π
2 2
π
l
1.2 Historical survey
Thus
Sn0 (α)
23
2n+1 ∗
n πl 2
−n
,
=
cos (l · α) l (n/2)
2 4
1 n (α + 2lπ )2
(2π )1/2 n−1
(α + 2lπ ) + n
− ,
2 (n/2)
2 2
4π
l
2π n/2
−
.
n(n/2)
(1.2.78)
The derivatives of this expression with respect to the components of α are easily
worked out with the aid of a recursion relation for the incomplete gamma function:
∂
(a, x) = a(a, x) − (a + 1, x).
(1.2.79)
∂x
The resulting sums converge reasonably rapidly. This procedure was adapted by
Heller and Marcus [75] to treat a class of dipole sums.
In 1952 Bertaut [11] derived and generalized Ewald’s summation formula by
the following procedure, which has not yet been fully exploited. We are interested
in calculating the electrostatic energy of a lattice of point charges q j located at
r j = Rl + xk , where the Rl form a Bravais lattice and the xk lie in the unit cell.
Denote the reciprocal lattice vectors by h. Then the electrostatic energy is
x
W = Wi + Ws ,
qi q j
1
Wi =
,
ri j = |ri − r j |,
2
ri j
(1.2.80)
i= j
and Ws is the (infinite) self-energy. By a well-known theorem in electrostatics,
W is unchanged if the point charges are replaced by non-overlapping spherical
charge densities:
σ (r ) dr = 1.
(1.2.81)
q j δ(r ) → q j σ (r ),
Then the charge density is
ρ(x) =
q j σ (x − r j ).
(1.2.82)
j
Let
φ(q) =
F(q) =
σ (x) e2πiq·x dx,
qk e2πiq·xk
(the structure factor).
k ∈ cell
Then
ρ(x) ρ(x + y)
dx dy
x
1 |F(h)|2
=
|φ(h)|2 ,
2π V
h2
1
W =
2
(1.2.83)
24
Lattice sums
where V is the volume of the unit cell. Alternatively,
σ (x − r j ) σ (x + y − ri )
1
W =
dx dy
qi q j
2
|x − r j |
i, j
1
du
=
qi q j dq |φ(q)|2
exp [2πiq · (u − ri j )]
2
u
i, j
sin (2πqri j )
1 qi q j ∞
dq
(1.2.84)
=
|φ(q)|2
π
ri j 0
q
i, j
since φ(q) is also spherically symmetric. Similarly, the self-energy is given by
letting ri j → 0:
∞
Ws = 2
qi2
|φ(q)|2 dq.
(1.2.85)
i
0
Hence the interaction energy is
Wi =
∞
1 |F(h)|2
2
|φ(h)|
−
2
qi2
|φ(q)|2 dq,
2π V
h2
0
(1.2.86)
i
h
from (1.2.83), and
Wi =
∞
sin (2πqri j )
1 qi q j ∞
dq − 2
|φ(q)|2
qi2
|φ(q)|2 dq
π
ri j 0
q
0
i, j
i
(1.2.87)
from (1.2.84).
Next suppose that σ (x) does not vanish outside the sphere whose radius is the
minimum intercharge spacing. Then a term A, the correction due to overlap, must
be added to the right-hand side of (1.2.86). However, from (1.2.84),
sin (2πqri j )
1 qi q j ∞
dq + A,
(1.2.88)
|φ(q)|2
W =
π
ri j 0
q
i, j
where
1 qi q j
2 ∞
2 sin (2πqri j )
A=
1−
dq .
|φ(q)|
2
ri j
π 0
q
i, j
Therefore we have
∞
1 qi q j
1 |F(h)|2
2
=
|φ(h)|
−
2
qi2
|φ(q)|2 dq
2
ri j
2π V
h2
0
i, j
i
h
∞
qi q j
sin (2πqri j )
2
1
1−
dq . (1.2.89)
+
|φ(q)|2
2
ri j
π 0
q
i, j
1.2 Historical survey
25
The only conditions on φ are
(1) φ(0) = 1;
(2) |φ(q)|2 is integrable on [0, ∞].
The potential at x = xk is
V (xk ) =
∞
∂ Wi
1 Sk
2
=
|φ(h)|
−
4q
|φ(q)|2 dq
k
∂qk
πV
h2
0
h
qj
2 ∞
2 sin 2πq|R j − xk |
1−
dq ,
+2
|φ(q)|
|R j − xk |
π 0
q
j
(1.2.90)
where
Sk =
qk exp [2πih · (xk − xk )].
(1.2.91)
k ∈ cell
For example, if we take φ(h) = exp (−π h 2 /2K 2 ), (1.2.90) is precisely Ewald’s
formula. Bertaut’s procedure was extended to multipole sums by Kanamori
et al. [93].
An equivalent procedure was developed and extensively applied by Nijboer
and de Wette [110], although actually it had been used several years earlier by
Placzek et al. [114]. The basic idea is that if S = n F(Rn ) converges slowly,
then one should select a rapidly converging function φ(R) which is well behaved
near R = 0 and write
F(Rn ) φ(Rn ) +
F(Rn )[1 − φ(Rn )].
(1.2.92)
S=
n
n
The first series now converges well and the second can be transformed into a
rapidly convergent form by Parseval’s theorem,
n ),
n ) G(K
F(Rn ) G(Rn ) =
(1.2.93)
F(K
n
n
denotes the Fourier transform of F. As an example this was applied to
where F
Slm
(0|k, n) =
ylm (θ, φ) exp (2πik · Rλ )
λ
Rλ2n+1
(1.2.94)
with
φ(R) =
(n + l, π R 2 )
(n + l)
(1.2.95)
26
Lattice sums
to obtain
Slm
(0|k, n) =
(n − l, π R 2 ) ylm (θ, φ) exp (2πik · Rλ )
1
λ
(n + l)
Rλ2n+1
λ
+
i l π 2n+l−3/2 |hλ − k|2n+l−3
λ
(−n + 32 , π |hλ − k|2 ) ylm (θ , φ ) .
(1.2.96)
Here (θ , φ ) are the polar angles of h − k and is the volume of the unit cell. In
this way Kanamori et al. [93] evaluated several multipole terms of the electrostatic
potential in a number of ionic lattices.
In an interesting note Nijboer and de Wette [111] discussed the conditionally
convergent series
P2 (cos θ )
(1.2.97)
Rλ3
λ
for a cubic lattice. This represents the field at the origin due to identical dipoles at
all other lattice points and, by the Lorenz argument, should have the value − 13 π .
However, if summed over successive shells the sum must vanish by symmetry. In
keeping with the Lorenz ‘parallel capacitor’ argument, Nijboer and de Wette proposed that the series should be summed first over planes of dipoles and then over
the various lattice planes. By separating out the summation over planes, which
leads to a geometric sum, they applied their method to the two-dimensional planar
sums and showed that, to computational accuracy, the value − 13 π is obtained. This
planewise summation method was applied to calculating electrostatic potentials in
metals by Sholl [126] and by McNeil [105]. A general discussion of the validity
of resumming such conditionally convergent series was given by Campbell [31].
This planewise procedure was carried out on similar sums for a variety of lattices
by de Wette [37] and de Wette and Schacher [38]. Nijboer’s and de Wette’s
scheme
√
was in turn generalized to the case of potentials f (R) for which f ( R)R −1/2 can
be represented as a Laplace transform by Adler [1], who made use of the earlier
work of Molière [107]. Adler gave explicit consideration to the sums
Ck f (|rkj |)
Am ylm (θ kj , φ kj ) exp (iK · rkj )
(1.2.98)
=
j,k
m
which were required for studies of various optical properties of solids (see also
Rudge [119]).
Beginning about 1950, Emersleben published an extensive series of papers
dealing with the electrostatic potentials of finite arrays of electric charges,
[42, 43], but we shall not describe this work here since it bears only tangentially on our subject. In addition, Emersleben devoted a number of papers to
various properties of Epstein zeta functions, [44, 45]. Since most of them deal
1.2 Historical survey
27
with functions of order 2, this work is subsumed and greatly generalized by more
recent work to be described below. Of particular interest, however, is the pair of
identities [45]
(A)
(B)
0
···
0
···
Z k1 =1
k P =1 h 1 + k1 /n . . . h P + k P /n
0
···
0 (s)
= n P−s Z n h , ..., n h 1
P
n
n
0
Z x
0
0 0 (s) + Z (s)
1
1
y
2 −x
2 −y
0
0 1−1/2s
(s).
=2
Z
x+y
x−y (s)
(1.2.99)
These are simply obtained as consequences of the cyclotomic equation (relating to sums over nth roots of unity) and represent analytic counterparts to the
Hund–Naor relations, discussed below, which were obtained by simple physical
arguments.
Another significant development during the 1950s was the series of papers by
Mackenzie [102], Benson [7], Benson and Schreiber [8], and Benson et al. [9]
dealing with the practical evaluation of electrostatic sums, which were reduced to
rapidly convergent series of modified Bessel or exponential functions. We shall
illustrate their method by the sum
S=
∞
[(k + a)2 + (l + b)2 ]−s .
(1.2.100)
k,l=−∞
The procedure is first to use (1.2.9) to exponentiate the summand, to apply
Jacobi’s transformation in the form
∞
p=−∞
e
−( p+α)2 t
=
∞
π 1/2 t
e−q
2 π 2 /t
cos (2qαπ ),
(1.2.101)
q=−∞
and then to perform the integral by means of Hobson’s representation for modified
Bessel functions of the second kind,
∞
2
2 2
t β−1 e−k t−q π /t dt = 2(qπ/k)β K β (2π kq).
(1.2.102)
0
28
Lattice sums
In this way we obtain the rapidly convergent series
∞
q s−1/2
K s−1/2 [2π |q(l + b)|] cos (2qπa)
l + b q,l=−∞
∞
2π s q s−1/2
K s−1/2 [2π |q(l + a)|] cos (2qπ b).
=
l + a (s)
2π s
S=
(s)
(1.2.103)
q,l=−∞
Van der Hoff and Benson [134] provided a number of such representations for
a variety of important electrostatic sums. It is interesting to note that, by applying this procedure to multiple sums whose values are known, the corresponding
Bessel function series are obtained in closed form, as has been formalized by Hautot [73]. For example, applying this procedure to the Lorenz–Hardy series (Hardy
[71]) discussed below, we obtain
∞
K 0 (mnπ ) = 14 [(4 −
√
2)ζ ( 12 )β( 12 ) + ln (8π ) − γ ].
(1.2.104)
m,n=1
The following application is instructive and furnishes a very concise representation for the Madelung energy of the rock salt structure.
We have
μ=−
l1
l2
(−1)l1 +l2 +l3 (l12 + l22 + l32 )−1/2 ,
(1.2.105)
l3
where the summation is over −∞ < li < ∞ and the prime denotes that l1 = l2 =
l3 = 0 is omitted. Next we write, by symmetry,
μ=−
l1
= −3
l2
(−1)l1 +l2 +l3 (l12 + l22 + l32 )(l12 + l22 + l32 )−3/2
l3
l1
l2
(−1)l1 l12 (l12 + l22 + l32 )−3/2 (−1)l2 +l3 ,
(1.2.106)
l3
and the prime can be omitted. By using (1.2.9) and (1.2.74) in the form
∞
(−1) exp (−l t) =
l
l=−∞
2
π 1/2 t
m odd
−m 2 π 2
,
exp
4t
we find that
∞
∞
12 l1 2
μ = −√
(−1) l1
exp (−l12 t) t 1/2 φ(t) dt,
π
0
l1 =1
(1.2.107)
1.2 Historical survey
where
φ(t) =
29
(−1)l2 +l3 exp [−(l22 + l32 )t]
l2
l3
−(l22 + l32 )π 2
π exp
=
.
t l l
4t
2
(1.2.108)
3
odd
Next we note that
∞
(l 2 + l32 )π 2 1/2
exp −l12 t − 2
t dt
4t
0
=2
π(l22 + l32 )1/2
2l1
1/2
K 1/2 [πl1 (l12 + l22 )1/2 ]
= π 1/2l1−1 exp [−πl1 (l22 + l32 )1/2 ]
(1.2.109)
so that
μ = −12π
∞ l1 =1 l2 ,l3
odd
l1 (−1)l1 exp [−πl1 (l22 + l32 )1/2 ].
(1.2.110)
Finally, summing the geometric l1 -series we obtain
μ = 12π
l2 ,l3
odd
= 12π
exp [−π(l22 + l32 )1/2 ]
{1 + exp [−π(l22 + l32 )1/2 ]}2
∞
sech2
π
2
(m 2 + n 2 )1/2 .
(1.2.111)
m,n=1
odd
This simple and rapidly convergent expression was discovered by Benson et al.
[9] using a physical argument and was proved analytically by Mackenzie [101].
A related representation was worked out for the CsCl structure by Benson and
van Zeggeren [10] and the original physical argument for (1.2.111) was extended
to all rhombohedral lattices by Mackenzie [102]. A number of these results were
obtained and generalized by Fumi and Tosi [53], who used Madelung’s method.
In 1959 Maradudin and Weiss [104] devised an efficient numerical scheme for
the sum
mn
,
(1.2.112)
Ts =
2
(m + n 2 )s/2
m,n
considered previously by Topping [132], which used the one-dimensional Poisson
summation formula and asymptotic approximations for the incomplete gamma
30
Lattice sums
functions. The procedure appears to be useful for similar higher-dimensional sums
but is too specialized for general application.
The two-dimensional Poisson summation formula was the implicit basis for
Madelung’s work in 1918 but was apparently not exploited more fully until
the mid 1950s, when it was used explicitly by Nicholson [109] and Hove and
Krumhansl [79] to rederive several older summations. We shall describe this procedure in more detail when we come to recent work by Pathria and Chaba [35],
who have employed it extensively.
Harris and Monkhorst [72] provided an interesting development of Nijboer’s
and de Wette’s approach. They treated a given lattice as if it were composed of
neutral cells made up of positive point charges embedded in a compensating uniform background of negative charge. By the use of Fourier integral representations
they were able to represent the electrostatic potential at a point as a sum of two
parts. One part depended on the lattice structure only and was a measure of the
potential at a lattice point when we have unit positive charges at all lattice points
balanced by negative charge distributed uniformly throughout the crystal. This
term gave the electrostatic energy of the low-density plasma model for that given
lattice. The second term depended both on the lattice structure and on the shape
of the compensating charge distribution. If this balancing charge was replaced by
localized point charges at appropriate points, this second term together with the
first gave the Madelung energy of the lattice. Redlack and Grindlay [115, 116]
in a series of papers considered the evaluation of Coulomb energies in finite size
crystals of different shapes. They too were able to divide the potential into an
intrinsic part, which was shape independent, and an extrinsic part, which was a
function of shape and size. Their numerical evidence suggests that the intrinsic
potential energy equals the Madelung energy of an infinite crystal.
We believe that this summarizes the state of affairs up to approximately 1973. It
is interesting to note that there was little attention given to the question of whether
these important lattice sums might be obtained exactly in closed form. Starting in
1973 this question began to be considered in earnest and we deal with it in the
next section.
1.3 The theta-function method in the analysis of lattice sums
Theta functions have, of course, been used in the numerical evaluation of lattice
sums since the work of Appell [2] and Ewald [49], and the Poisson summation
formula so often invoked is just a particular case of a θ -function transformation.
However, the analytic power of representing lattice sums as Mellin transforms
of θ -functions (MTθ Fs) does not appear to have been utilized until 1973 when
Glasser [55], [56] rediscovered and extended this approach. In order to understand the method, properties of θ -functions relevant to lattice sums will now be
described.
1.3 The theta-function method in the analysis of lattice sums
31
Theta functions were first introduced by Jacobi [82] as a means of calculating
elliptic functions. They are functions of a complex variable z and a parameter q.
Our interest is in the case z = 0, so that for simplicity we consider θ -functions
to depend on the single variable q. These functions have remarkable properties in
that they may have both infinite series and infinite product representations; this
leads to some astonishing identities among sums and products of θ -functions.
Following Whittaker and Watson [139] we define the following quantities:
θ2 =
θ3 =
θ4 =
∞
q (n−1/2) = 2q 1/4 (1 + q 2 + q 6 + · · · + q n(n+1) + · · · ), (1.3.1)
2
−∞
∞
−∞
∞
2
q n = 1 + 2q + 2q 2 + 2q 9 + · · · ,
(1.3.2)
2
(−1)n q n = 1 − 2q + 2q 2 − 2q 9 + · · · ,
−∞
∞
θ1 = 2
(−1)n (2n + 1)q (n+1/2)
(1.3.3)
2
0
= 2q 1/4 (1 − 3q 2 + 5q 6 − 7q 12 + · · · ).
(1.3.4)
To these we add two series not given by Jacobi,
θ5 = 2
∞
(−1)n q (2n−1/2) = 2q 1/4 (1 − q 2 − q 6 + q 12 + q 20 + · · · ), (1.3.5)
2
−∞
θ6 = 2q 1/4 (1 + 3q 2 − 5q 6 − 7q 12 + · · · ).
(1.3.6)
All these series can be expressed in terms of the infinite products
Q0 = ∞
(1 − q 2n ),
Q1 = ∞
(1 + q 2n ),
1∞
1∞
2n−1
),
Q 3 = 1 (1 − q 2n−1 ),
Q 2 = 1 (1 + q
(1.3.7)
viz.,
θ2 = 2q 1/4 Q 0 Q 21 ,
θ3 = Q 0 Q 22 ,
θ5 = 2q 1/4 Q 0 Q 1 /Q 2 (q 2 ),
θ4 = Q 0 Q 23 ,
θ1 = 2q 1/4 Q 30 ,
θ6 = 2q 1/4 Q 30 (q 2 )Q 32 (q 2 ).
(1.3.8)
It is thus a matter of elementary algebra to obtain a large number of relations
among the θ ’s; we list just a few to show the variety available. We have the
additive relations
θ3 + θ4 = 2θ3 (q 4 ),
θ3 − θ4 = 2θ2 (q 4 )
θ32 + θ42 = 2θ32 (q 2 ),
θ32 − θ42 = 2θ22 (q 2 )
θ34 = θ44 + θ24 .
(1.3.9)
32
Lattice sums
This last equation written in infinite product form,
∞
8
(1 + q 2n−1 )
=
1
∞
8
(1 − q 2n−1 )
+ 16q
1
∞
8
(1 + q 2n )
, (1.3.10)
1
led Jacobi [82] to describe it as ‘aequatio identica satis abstrusa’. It is easy to show
that Q 1 Q 2 Q 3 = 1, Q 0 Q 3 = Q 0 (q 1/2 ), Q 1 Q 2 = Q(q 1/2 ), Q 2 Q 3 = Q 3 (q 2 ),
Q 0 Q 1 = Q 0 (q 2 ), and these lead to multiplicative relations among the θ ’s, some
examples of which are
θ3 θ4 = θ42 (q 2 ),
2θ2 θ3 = θ22 (q 1/2 ),
2θ2 θ4 = θ52 (q 1/2 ),
θ1
θ2 θ4 (q 4 ) = θ5 θ3 (q 2 ),
θ6
= θ2 θ3 θ4 ,
=
θ5 θ32 (q 2 ).
(1.3.11)
(1.3.12)
It is in these last two results that the power of θ -functions is best shown – namely
the ability to reduce a product of multiple sums (in this case three) to just one
sum. Nor are these the only examples. Jacobi gives many others by which squares,
fourth powers, sixth powers, and even eighth powers of θ -functions may be contracted into single sums. Indeed, in certain combinations, any even power of
θ -functions may be so contracted. Two examples are
θ32
=1+4
∞
1
qn
,
1 + q 2n
θ48
∞
n3 q n
= 1 + 16
(−1)n
.
1 − qn
(1.3.13)
1
A fairly large list of such relations was given by Zucker [142]. We now illustrate
how to obtain lattice sums analytically using such relations. First define the Mellin
transform Ms of f (t) by
∞
(s)Ms [ f (t)] =
t s−1 f (t) dt.
0
Let
S(a, b, c) =
(al12 + bl1 l2 + cl22 )−s
(l1 ,l2 =0,0)
be a general two-dimensional sum. Then, e.g.,
S(1, 0, 1) =
(l12 + l22 )−s = Ms
exp [−(l12 + l22 )t] = Ms [θ32 − 1]
(q = e−t ).
Now, by using
θ32 − 1 = 4
∞
1
∞
∞
1
0
qn
=4
q n (−1)m q 2nm ,
2n
1+q
1.3 The theta-function method in the analysis of lattice sums
33
we have
Ms [θ32 − 1] =
4
(s)
=4
∞
∞
t s−1
0
n −s
n=1
∞ ∞
(−1)m exp [−n(2m + 1)t] dt
n=1 m=0
∞
(−1)m (2m + 1)−s = 4 ζ (s) β(s),
(1.3.14)
m=0
where
β(s) =
∞
(−1)m (2m + 1)−s .
0
Thus, a two-dimensional sum has been evaluated exactly, i.e., we have expressed
it as the product of simple sums. Such a result was first obtained by Lorenz [100].
Four steps are taken in this approach:
(1)
(2)
(3)
(4)
the sum is written as a Mellin transform;
the transformed sum is written in terms of θ -functions of argument q = e−t ;
the θ -functions series are transformed to other q-series where possible;
the Mellin transform of the new series is re-evaluated.
We give another illustration with the eight-dimensional sum
(−1)l1 +l2 +···+l8 (l12 + l22 + · · · + l82 )−s
=
Ms (θ48
− 1) = Ms 16
∞
(−1)n n 3 q n
1
= Ms 16
∞ ∞
1
η(s) =
1 − qn
(−1)n n 3 q n q nm = −16 ζ (s) η(s − 3),
(1.3.15)
1
∞
(−1)n+1 n −s .
1
When s = 4 this yields the attractive result that
8π 4 ln 2
.
(1.3.16)
45
Many similar examples are given by Glasser [56, 57] and Zucker [142]. The essential point of the four steps is whether we are able to find suitable expressions for
the θ -function representation of a lattice sum. We can write quite generally
(m 2 + λn 2 )−s = Ms [θ3 θ3 (q λ ) − 1],
(1.3.17)
S(1, 0, λ) =
(−1)l1 +l2 +···+l8 (l12 + l22 + · · · + l82 )−4 = −
but can proceed no further unless we have a q-series for θ3 θ3 (q λ ). Rather
remarkably, many such series are available or may be constructed. If not, it
34
Lattice sums
appears that in other cases S(1, 0, λ) may be summed by number-theoretic techniques – see Section 1.4. This immediately poses the questions when such
series may be constructed and whether it is possible to say when some general
S(a, b, c) = (am 2 + bmn + cn 2 )−s may be summed exactly. It seems that,
for two-dimensional sums, a criterion for solvability is available in the form of
a conjecture by Glasser [57] as to when a two-dimensional sum is expressible in
products of simple sums. This has been verified in all the cases known (Zucker
and Robertson [147, 148]) and is discussed in more detail in Section 1.4.
Here we shall focus attention on the physically more interesting threedimensional sums. Now, whereas in two dimensions it appears that we know
precisely when we can express a lattice sum exactly, and also in four dimensions
where a vast number of exact results are obtainable, in contrast, in three dimensions there is a paucity of exact results. Indeed, only two are known through the
use of (1.3.12):
(−1)m [m 2 + n 2 + ( p − 12 )2 ]−s = Ms [θ2 θ3 θ4 ] = Ms [θ1 ]
= 22s+1 β(2s − 1),
(1.3.18)
m
2
2 −s
2 2
1 2
(−1) [(2m − 2 ) + 2n + 2 p ] = Ms [θ5 θ3 (q )] = Ms [θ6 ]
= 22s+1 L −8 (2s − 1),
(1.3.19)
where L −8 (s) = 1 + 3−s − 5−s − 7−s · · · is an L-function (see Section 1.4). Neither of these results is of immediate physical interest; however, though θ -functions
have not yet provided any physically important exact results in three dimensions,
they still yield concise representations of lattice sums and their manipulation
does away with the necessity of resorting to geometrical methods of handling
three-dimensional sums. This process is seen at its best when applied to Hund’s
procedure described above for calculating lattice potentials. By taking any lattice
site as the origin of a coordinate system, Hund [81] set out to calculate the interaction of the origin particle with all other particles in a crystal. If there are several
species of particle, each species being defined by a Bravais lattice spanned by
the basis vectors a1 (i), a2 (i), a3 (i), one can place a particle of each species in
turn at the origin and calculate its interaction with all other particles in terms of
some characteristic length a, where, e.g., a = |a1 (i)|. This pure number is called
the Hund potential of the ith species. By adding all the Hund potentials together
and dividing by 2 (to allow for the fact that every interaction is counted twice)
the total interaction energy of the crystal is expressed as a pure number in terms
of the characteristic length. Hund was mainly concerned with ionic crystals for
which the interaction is Coulombic, but his method is easily generalized.
Let the coordinates of the nearest particle of species i to the origin be xa1 ,
ya2 , za3 , where x, y, z < 1. Let the origin particle interact with this species via
a potential function F(r ), where r is given by r 2 = (m − x)2 a21 + (n − y)2 a22 +
( p −z)2 a23 +2(m − x)(n − y)a1 ·a2 +2(m − x)( p −z)a1 ·a3 +2(n − y)( p −z)a2 ·a3
1.3 The theta-function method in the analysis of lattice sums
35
and m, n, p are integers taking all values from −∞ to ∞ independently of one
another. Then the Hund potential of the origin particle with this species is
r a12 a22 a32 2a1 · a2 2a2 · a3 2a1 · a3
.
,
,
,
,
,
,
x,
y,
z
:
F
=
F
a
a2 a2 a2
a2
a2
a2
m,n, p
(1.3.20)
The interaction of the origin particle with its own species corresponds to x = y =
z = 0, and in the summation the point m = n = p = 0, which gives the interaction of the origin particle with itself, is excluded. It is quite straightforward to
evaluate H for any structure and potential function. However, many simplifications may be made in cases of physical interest. Thus all species will generally be
described by the same basic vectors a1 , a2 , a3 .
Hund, in fact, concentrated on hexagonal and cubic structures. In the hexagonal
case all lattice sites may be specified in terms of |a1 | = |a2 | = a, 2a1 · a2 = a 2 ,
a1 · a3 = a2 · a3 = 0, and |a3 | = ca, where c is called the axial ratio. For such
crystals Hund evaluated some potentials for F(r ) = r −l , namely
H
H (1, 1, 83 , 1, 0, 0, x, y, z : r −1 ) = φ(x, y, z)
(1.3.21)
H (1, 1, 4, 1, 0, 0, x, y, z : r −1 ) = X (x, y, z).
(1.3.22)
and
For cubic structures |a1 | = |a2 | = |a3 | = a; a1 · a2 = a2 · a3 = a3 · a1 = 0 and
the Hund cubic potential is
H (1, 1, 1, 0, 0, 0, x, y, z : r −1 ) = ψ(x, y, z : r −1 ).
(1.3.23)
Since most typical ionic crystals have cubic structures, we will concentrate on
them. Thus ψ for particles interacting with an r −s potential may be written
ψ(x, y, z : r −s ) =
∞ = Ms
−∞
∞
−∞
q
[(m − x)2 + (n − y)2 + ( p − z)2 ]−s
(m−x)2
∞
−∞
q
(n−y)2
∞
q ( p−z)
2
(1.3.24)
−∞
and, if x, y, z take on special values, we may express ψ in terms of θ -functions.
Naor [108], in a little known but very important paper, fully developed Hund’s
method to make a comprehensive study of cubic structures without free parameters. These cubic structures are those whose lattice sites lie at the intersection point
of symmetry elements and are termed invariant cubic lattice complexes (ICLCs).
For such structures (x, y, z) can only take on the values zero or multiples of 18 .
Further, to preserve symmetry, a species may be required to occupy a particular set
of sites simultaneously. Naor [108] then showed that only 17 potentials, which he
denoted by capital roman letters, were required to calculate μ for all ICLCs. These
36
Lattice sums
Naor potentials were all various combinations of Hund potentials (see Table 1.1).
By making use of identities determined by Hund from geometric considerations
(see (1.2.67)–(1.2.69)), Naor found that only nine of these potentials were independent and conjectured that a linear combination of nine independent lattice
sums was both sufficient and necessary to give the Madelung constant of any
ionic crystal that was an ICLC. However, expressing ψ in terms of MTθ Fs not
only makes the Hund identities obvious but furthermore shows that a hitherto
unknown relation exists between two Naor potentials.
∞ (m−x)2
, which appears in the
For example, consider the sum s(x) =
−∞ q
MTθ F representation of ψ. We have the following pairs of values:
x = 0, s(0) = θ3 ;
x = 12 , s( 12 ) = θ2 ;
x = 14 , s( 14 ) = 2−1 θ2 (q 1/4 );
x = 18 , s( 18 ) = 2−2 [θ2 (q 1/16 ) + θ5 (q 1/16 )];
x = 38 , s( 38 ) = 2−2 [θ2 (q 1/16 ) − θ5 (q 1/16 )].
(1.3.25)
=
may be
Furthermore s(x) = s(l − x). Hence
expressed entirely in combinations of the θ -functions (1.3.1)–(1.3.6), and the full
array of identities among them may be used to generate relations among the ψ.
For example, ψ(0, 0, 0 : r −2s ) = Ms [θ33 − 1], where the 1 is subtracted in the
transform to account for the omission of the term m = n = p = 0 in the original
sum. Furthermore,
ψ( 18 q , 18 q , 18 q )
ψ(0, 0, 12 : r −2s ) = Ms [θ32 θ2 ];
ψ( 18 q )
ψ(0, 12 , 12 : r −2s ) = Ms [θ3 θ22 ];
ψ( 12 , 12 , 12 : r −2s ) = Ms [θ23 ].
Now using the result θ3 = θ3 (q 4 ) + θ2 (q 4 ) plus the important property of Mellin
transforms that
Ms [ f (q k )] = k −s Ms [ f ],
(1.3.26)
we have
ψ(0, 0, 0 : r −2s ) = Ms [θ33 − 1] = Ms {[θ3 (q 4 ) + θ2 (q 4 )]3 − 1}
= 2−2s Ms [(θ3 + θ2 )3 − 1]
= 2−2s Ms [θ33 − 1 + 3θ32 θ2 + 3θ3 θ22 + θ23 ]
= 2−2s [ψ(0, 0, 0 : r −2s ) + 3ψ(0, 0, 12 : r −2s )
+ 3ψ(0, 12 , 12 : r −2s ) + ψ( 12 , 12 , 12 : r −2s )],
(1.3.27)
and when s = 12 we have the Hund identity (1.2.69) without recourse to
geometrical methods. Again,
ψ( 14 , 14 , 14 : r −2s ) = 2−3 Ms [θ23 (q 1/4 )] = 22s−3 Ms [θ23 ]
= 22s−3 ψ( 12 , 12 , 12 : r −2s )
Table 1.1 Naor and Hund potentials for the cubic lattices (k = 22s−3 )a
Naor
symbol
Hund potential
Theta function representation
Value for general s
Value for s = 12
Numerical value
a = k(a + 3b + 3c + d)
a(1) = b(1) + c(1) + d(1)/3
−2.837297479480619
A
ψ(0, 0, 0)
θ33 − 8 = k[(θ3 + θ4 )3 − 1]
B
ψ(0, 12 , 12 )
ψ(0, 0, 12 )
ψ( 12 , 12 , 12 )
ψ( 14 , 14 , 14 )
ψ( 14 , 14 , 12 )
1 [ψ(0, 0, 1 )
2
4
+ ψ( 14 , 14 , 12 )]
ψ(0, 14 , 14 )
ψ(0, 14 , 12 )
1 [ψ( 1 , 1 , 1 )
4
8 8 8
+ 3ψ( 18 , 38 , 38 )]
1 [ψ( 3 , 3 , 3 )
4
8 8 8
+ 3ψ( 18 , 18 , 38 )]
1 [ψ(0, 1 , 1 )
2
8 8
+ ψ( 12 , 38 , 38 )]
θ3 θ22 = k[(θ3 + θ4 )(θ3 − θ4 )2 ]
θ32 θ2 = k[(θ3 + θ4 )2 (θ3 − θ4 )]
θ23 = k(θ3 − θ4 )3
2−3 θ23 (q 1/4 ) = k 2 (θ3 − θ4 )3
k[θ22 (θ3 − θ4 )]
G
D
E
J
K
Y
Z
L
M
N
k(a − b − c + d)
d(1)/3
−0.582521531544394
k(a + b − c − d)
[3b(1) − d(1)]/6
−0.095932304939804
k(a − 3b + 3c − d)
[3c(1) − a(1)]/2
−0.801935970028024
k 2 (a − 3b + 3c − d)
[3c(1) − a(1)]/8
−0.200483992507006
k 2 (a − b − c + d) − ke
[d(1) − 3e(1)]/12
+0.305934570506908
k[θ2 (θ32 + θ42 )]
k 2 (a + b − c − d) + k f
[3b(1) − d(1) + 6 f (1)]/24
−0.530673108834998
k[θ22 (θ3 + θ4 )]
k 2 (a − b − c + d) + ke
[d(1) + 3e(1)]/12
+0.239412342577787
k[θ2 (θ32 − θ42 )]
k 2 (a + b − c − d) − k f
[3b(1) − d(1) − 6 f (1)]/24
−0.353900722976810
k 2 [θ23 + θ53 ]
k 3 (a − 3b + 3c − d) + k 2 g
[3c(1) − a(1) + 2g(1)]/32
+0.108226339650319
k 2 [θ23 − θ53 ]
k 3 (a − 3b + 3c − d) − k 2 g
[3c(1) − a(1) − 2g(1)]/32
−0.208468335903822
k 2 [(θ3 + θ4 )(θ22 + θ52 )]
+ 22 k[θ2 θ3 θ4 ]
k 3 (a − b − c + d) + k 2 (e + h + j)
[d(1) + 3e(1) + 3h(1)
+ 25s−2 β(2s − 1)
+ 3 j (1)]/48 + 2−1/2
+1.074506464381240
Table 1.1 (cont.)
Naor
symbol
P
R
S
X
T
Value for s = 12
Hund potential
Theta function representation
Value for general s
1 [ψ( 1 , 1 , 1 )
2
2 2 8
+ ψ(0, 38 , 38 )]
1 [ψ( 1 , 1 , 1 )
2
4 8 8
+ ψ( 14 , 38 , 38 )]
1 [ψ(0, 1 , 3 )
2
8 8
+ ψ( 12 , 18 , 38 )]
ψ( 18 , 14 , 38 )
k 2 [(θ3 + θ4 )(θ22 + θ52 )]
k 3 (a − b − c + d) + k 2 (e + h + j)
[d(1) + 3e(1) + 3h(1)
− 22 k[θ2 θ3 θ4 ]
k 2 [(θ3 − θ4 )(θ22 + θ52 )]
− 25s−2 β(2s − 1)
− 3 j (1)]/48 − 2−1/2
1 [ψ(0, 1 , 1 )
4
8 4
+ ψ(0, 14 , 38 )
+ ψ( 12 , 18 , 14 )
+ ψ( 12 , 14 , 38 )]
k 2 [(θ3 + θ4 )(θ22 − θ52 )]
k 3 (a − b − c + d)
[d(1) − 3e(1) + 3h(1)
+ k 2 (−e + h − j)
− 3 j (1)]/48
k 3 (a − b − c + d) + k 3 (e − h − j)
k 2 [θ2 (θ32 − θ42 )]
−0.339707097991855
+0.014147501743402
[d(1) + 3e(1) − 3h(1)
−3 j (1)]/48
k 2 [(θ3 − θ4 )(θ22 − θ52 )]
Numerical value
−0.247693511905799
k 3 (a − b − c + d)
[d(1) − 3e(1) − 3h(1)
+ k 2 (−e − h + j)
+ 3 j (1)]/48
−0.279484055918394
k 3 (a + b − c − d) − k 3 f
[3b(1) − d(1) − 6 f (1)]/96
−0.088475180744203
a Hund’s identities: A = 3(B + G) + D, B = 2(Y + J ), D = 4E, E = 2(L + M), G = 2(Z + K ), J = 2(X + R), Y = N + P + 2S, Z = 4T, N − P = 22s−1 β(2s − 1).
1.3 The theta-function method in the analysis of lattice sums
39
and when s = 12 , 4ψ( 14 , 14 , 14 : r −1 ) = ψ( 12 , 12 , 12 : r −1 ), which is another of
Hund’s identities. Using this approach Naor’s 17 potentials may be expressed
in terms of MTθ Fs (Table 1.1). The Naor relations among the potentials are
immediately apparent and the further relation
N − P = 23s−2 Ms [θ1 ] = 25s−1 β(2s − 1)
(1.3.28)
√
may be discovered. Thus for s = 12 and with β(0) = 12 we have N − P = 2,
and only eight independent lattice sums at most are required to calculate μ for
all ICLCs. Sakamoto [122] had noted this result from numerical work but did not
give a proof. One cannot preclude the possibility that still further relations exist.
The eight independent (as far as is known) sums chosen here to calculate μ are:
(−1)m (m 2 + n 2 + p 2 )−s = Ms [θ4 θ32 − 1],
b = b(2s) =
(−1)m+n (m 2 + n 2 + p 2 )−s = Ms [θ42 θ3 − 1],
c = c(2s) =
d = d(2s) =
(−1)m+n+ p (m 2 + n 2 + p 2 )−s = Ms [θ43 − 1],
(−1)m [m 2 + (n − 12 )2 + ( p − 12 )2 ]−s = Ms [θ4 θ22 ],
e = e(2s) =
(−1)m+n [m 2 + n 2 + ( p − 12 )2 ]−s = Ms [θ42 θ2 ],
(1.3.29)
f = f (2s) =
g = g(2s) =
(−1)m+n+ p [(2m − 12 )2 + (2n − 12 )2 + (2 p − 12 )2 ]−s
= Ms [θ53 ],
(−1)n+ p [m 2 + (2n − 12 )2 + (2 p − 12 )2 ]−s = Ms [θ3 θ52 ],
h = h(2s) =
j = j (2s) =
(−1)m+n+ p [m 2 + (2n − 12 )2 + (2 p − 12 )2 ]−s = Ms [θ4 θ52 ].
This choice is governed by the fact that each sum is convergent (albeit conditionally) for all s > 0, whereas some Hund potentials expressed as sums are divergent.
For example, it is often convenient to use the self potential ψ(0, 0, 0 : r −2s ) =
2
(m + n 2 + p 2 )−s = Ms [θ33 − 1]. In its sum form a(2s) is
a = a(2s) =
divergent for s < 32 . But a(2s), which may be regarded as the three-dimensional
analogue of ζ (2s), has only a simple pole at s = 32 and may be analytically continued for s < 32 by various functional relations. One such relation, which again
uses θ -identities, is
a(2s) = Ms [θ33 − 1] = Ms [ 18 (θ3 (q 1/4 ) + θ4 (q 1/4 ))3 − 1]
= 22s−3 Ms [(θ3 + θ4 )3 − 8].
Hence
a(2s) = 22s−3 [a(2s) + 3b(2s) + 3c(2s) + d(2s)]
(1.3.30)
and, in particular, a(1) = b(1) + c(1) + d(1)/3. Since b(1), c(l), d(l) are all
convergent, we may give a value to a(1). Before considering these functional
40
Lattice sums
relations further it is instructive to show how the Hund–Naor method may be
used for finding μ. As examples we will find μ for the NaCl, CsCl, and CuF2
structures.
It is convenient to describe a species of particle in a cubic structure whose basic
coordinates are (x, y, z) as a simple cubic lattice based on (x, y, z), SC(x, y, z)
for short. Very often in cubic structures we find the same species occupying the
four sites (x, y, z), (x, y + 12 , z + 12 ), (x + 12 , y, z + 12 ), (x + 12 , y + 12 , z); that is,
the species forms a face-centred cubic lattice based on (x, y, z), FCC(x, y, z) for
short. Many crystals are easily described in these terms; thus NaCl with Na+ as
an origin may be described as Na+ FCC(0, 0, 0) + Cl− FCC( 12 , 12 , 12 ). Therefore,
for Coulomb interactions, the potential for Na+ is
ψ(Na+ ) = ψ(0, 0, 0) + 3ψ(0, 12 , 12 ) − ψ( 12 , 12 , 12 ) − 3ψ(0, 0, 12 )
and from Table 1.2 this is just 2d(1). If we now make Cl− the origin, the crystal is Cl− FCC(0, 0, 0) + Na+ FCC(0, 0, 0) and ψ(Cl− ) is also 2d(1). The total
Coulomb energy is thus 12 [ψ(Na+ ) + ψ(Cl− )] = 2d(1), and this is μ(NaCl)
by definition. Similarly, for CsCl, which is described by Cs+ SC(0, 0, 0) +
Cl− SC( 12 , 12 , 12 ), we have μ(CsCl) = 32 b(1) + 12 d(1). The case of CaF2 is
more complex. With Ca2+ as origin it is described by Ca2+ FC(0, 0, 0) +
F− FCC( 14 , 14 , 14 ) + F− FCC( 34 , 34 , 34 ) and, therefore,
ψ(Ca2+ ) = 4[ψ(0, 0, 0) + 3ψ(0, 12 , 12 )] − 16ψ( 14 , 14 , 14 )
= 6a(1) − 6c(1) + 4d(1).
With F− as an origin, CaF2 is F− FCC(0, 0, 0) + F− FCC( 12 , 12 , 12 ) +
Ca2+ FC( 14 , 14 , 14 ); thus
ψ(F− ) = ψ(0, 0, 0) + 3ψ(0, 12 , 12 ) + ψ( 12 , 12 , 12 ) + 3ψ(0, 0, 12 )
−2 × 4ψ( 14 , 14 , 14 )
= 2a(1) − 3c(1).
Hence
μ(CaF2 ) = 12 [ψ(Ca2+ ) + 2ψ(F− )] = 6a(1) − 6c(1) + 2d(1)
= 6b(1) + 4d(1).
It is immediately evident that
μ(CaF2 ) = 4μ(CsCl) + μ(NaCl).
(1.3.31)
This result or its equivalent was given first by P. Naor (unpublished) and by
Bertaut [12], in 1954, and was derived in the above manner by Fumi and Tosi
[52]. In a similar way it was shown that μ(CuO2 ) = 3b(1) + 3c(1) + 2d(1), and
so, unless there is some unknown relation among b, c, and d, μ(CuO2 ) cannot be
expressed in terms of μ(CsCl) and μ(NaCl). It is apparent, however, that because
1.3 The theta-function method in the analysis of lattice sums
41
Table 1.2 Madelung constants for a number of ionic crystals
Crystal
b(1)
c(1)
NaCl
d(1)
e(1)
g(1)
μ
2
3.495129189
CsCl
3/2
1/2
2.035361510
ZnS
3/2
3/2
CaF2
6
4
11.63657523
Cu2 O
3
3
2
10.25945703
6
2
12.37746803
9/2
44.55497526
KZnF3
3/2
24
(A R = 1)
√
(A R = 2)
1
1
(A R = 2)
1
1/2
6
24
24
8
49.50987213
3/2
24
25/2
58.53549202
LaAlO3
CaWO4
3.782926104
ReO3
2−1/2
BaTiO3
NaTaO3
2.254775948
2−1/2
2.282508448
−1/2
2.284666496
18
71.63183493
YOF
27/2
19/2
27.05607656
BiF3
6
10
22.12196280
1/2
10.91770035
BaLiF3
3/2
NbO
1/2
Li2 O
3/2
PdO
(AR =
√
2)
10.77318449
1/2
3/2
6
6
[Pt(0, 0, 0)S( 41 , 14 , 14 )]
9/8
3/4
3/8
2.636813487
[Pt(0, 0, 0)S( 21 , 0, 0)]
1/2
1
1/2
2.741365176
[Pt(0, 0, 0)S( 21 , 0, 12 )]
1
3.649338711
1
1
−1/2
4.539442445
8
27.50917245
2.254775948
(CaFC(0, 0, 0),
F I FC( 14 , 14 , 14 ),
F I I FC( 21 , 0, 0))
3/2
11/2
10.77318449
63.49038889
K2 PtCl6
24
16
(NH4 )3 AlF6
24
10
6
33/2
75/2
CuVS4
Spinels
3.008539964
11/2
2
ZrI4
CaF2
3/2
3/2
(AR = 2)
PtS
6
53.00500132
−6
66.7566350
Normal
p = 2, q = 3
p = 3, q = 5/2
p = 4, q = 2
3
3
61
−6
128.5671125
27/4
27/4
221/4
−15/2
130.7743620
12
12
52
−8
138.1991295
Inverse
p = 4, q = 2
4
4
62
−6
132.5694531
p = 2, q = 3
7
7
111/2
−15/2
131.7749472
42
Lattice sums
of the identities existing among Hund potentials, the number of independent sums
required to obtain μ for different structures based on the same fundamental lattice will be small and we should not be surprised that relations such as (1.3.31)
exist. What is surprising is that such relations were not discovered until 1954.
Naor’s method was very complicated and used purely geometric techniques, by
which a given structure was decomposed into a superposition of other structures,
while Bertaut obtained his results by a procedure involving sums over the reciprocal lattice. As Tosi [133] points out, the Hund technique is straightforward and
of general applicability whereas the other methods are liable to error. Indeed,
Hund pointed out that Emersleben’s calculation of μ(Cu2 O) (Section 1.2) in Born
potentials is incorrect, whereas it may be found in a few lines in Hund potentials.
It is seen here that the sums b(1), c(1), and d(1) appear very often and, indeed,
for the vast majority of the common ionic crystals only these three numbers are
required to give μ. In Table 1.2 a list of such values is given. That just three numbers were sufficient was noted by Sakamoto [120], who gave tables of μ for many
crystals in terms of three independent Born potentials. It is easy to transform from
Born to Hund potentials and back again via the relation
nπ
p
n
=
n−1 n−1 n−1
q1 =q2 =q3 =0
ψ
q
n
exp
−2πi(p · q)
,
n
(1.3.32)
but using Hund potentials to find μ is a simpler process. Sakamoto and his coworkers between 1953 and 1974 did a considerable amount of work on ionic
crystals and it is worth summarizing the main points of their results here. Takahasi
and Sakamoto [129] took up the problem of evaluating ψ to great accuracy, for
many Naor potentials were unknown. Hund [81] had obtained a few to about
three significant figures and Birman [15] added a few more. However, Takahasi
and Sakamoto give ψ(q/4), 0 ≤ qi ≤ 2, to 15 places and ψ(q/6), 0 ≤ qi ≤ 3, to
six places. Takahasi and Sakamoto also gave tables of ψ(q/24), 0 ≤ qi ≤ 12, to
at least five places, and Sakamoto [121] in one of his papers on μ for crystals with
quasi-cubic structures gave some values of ψ(q/n) with n = 960 and 1000 for
several values of qi which he required. These values are, of course, unnecessary
for ICLCs but are useful for crystals with cubic structures having free parameters.
Sakamoto has also privately communicated to us ψ(q/8), 0 ≤ qi ≤ 4, to 15
places and thus has computed Naor’s 17 potentials for ICLC to this accuracy (see
Table 1.1). All this numerical work was carried out by the Ewald method, which,
though effective, is a comparatively slow method for computing lattice sums for
Coulomb potentials. This method is still commonly used by workers evaluating
lattice sums but more powerful methods are available, such as that of Van der Hoff
and Benson [134] described above. (See also Graovac et al. [65] for an example
of a different sort of sum.) The latter method was extended by Hautot [73], [74]
to yield formulae for evaluating the better-known sums such as d(1) in terms of
1.3 The theta-function method in the analysis of lattice sums
43
elementary functions. So powerful are these techniques that these sums may be
evaluated on a hand machine to many decimal places in a few minutes. The Van
der Hoff–Benson–Hautot method makes essential use of the identities
∞
∞ 25/2−s π s 2m + 1 s−1/2
(−1)n (m 2 + b2 )−s =
(s)
b
−∞
m=0
× K s−1/2 [(2m + 1)bπ ],
∞
(m 2 + b2 )−s =
−∞
π 1/2 (s −
b2s−1 (s)
1
2)
+ 25/2−s
s ≥ 0,
∞ π s 2m s−1/2
(s)
m=1
× K s−1/2 [2mbπ ],
b
s > 0,
(1.3.33)
where K s is a modified Bessel function of the second kind (Watson [137]). Van
der Hoff and Benson derived their results by putting the summand into integral
form and, using the Poisson transformation formula
∞
∞
π 1/2 2
2 2
e−n t =
e−n π /t ,
(1.3.34)
t
−∞
−∞
transformed the sum into an integral representation for K s . Hautot used
Schlomilch series with a Hankel transform to obtain equivalent results. The
expansions are very rapidly convergent because the asymptotic behaviour of
K s (z) is (π/2z)1/2 exp (−z). Since many two-dimensional sums are known
exactly, (1.3.33) and (1.3.34) lead to interesting expressions for certain series of
modified Bessel functions. For example, if in (1.3.33) we put b = n and sum over
n, we have
∞ ∞ 25/2−s π s 2m + 1 s−1/2
n
2
2 −s
(−1) (m + n ) =
(s)
n
−∞
m=0
× K s−1/2 [(2m + 1)nπ ]
= −4 × 2−s η(s) β(s),
η(s) = (1 − 21−s )ζ (s).
In particular, if s = 12 , it can be established that
∞ ∞
m=1 n=1
√
ln 2 2 − 2 1
+
η( 2 ) β( 12 ).
K 0 [(2m + 1)nπ ] =
4
4
(1.3.35)
Many similar results may be obtained (Chaba and Pathria [35]; Hautot [74]) and
some are collected in Table 1.3. The application of (1.3.33) to evaluate d(1)
where b2 = n 2 + p 2 led Hautot to the formula
d(1) = 2 ln 2 + 4
(−1)m+n K 0 [π(n 2 + p 2 )1/2 (2m + 1)], (1.3.36)
which is essentially the result that Madelung [103] first obtained. However, no
geometric considerations such as those described in Section 1.2 are required.
44
Lattice sums
Table 1.3 Two-dimensional Bessel function sums
∞
K 0 (klπ ) =
k,l=1
odd
∞
∞
p=2
even
l=1
odd
∞ ∞
√
4−3 2 1
ζ ( 2 ) β( 12 )
8
K 0 ( plπ ) =
(−1)m−1 K 0 (mlπ ) =
m=1 l=1
odd
∞ ∞
K 0 (mlπ ) =
m=1 l=1
odd
∞
∞ K 0 (mpπ ) =
m=1 p=2
even
∞
K 0 (qpπ ), =
q, p=2
even
∞
K 0 (mnπ ) =
m,n=1
∞
m,n=1
∞
K 0 (2π mn) =
K 0 (4π mn) =
m,n=1
∞ ∞
m=1 n=1
√
2 1
1
ln 2 +
ζ ( 2 ) β( 12 )
4
8
√
1− 2 1
1
ζ ( 2 ) β( 12 ) − ln 2
2
4
√
2− 2 1
1
ln 2 +
ζ ( 2 ) β( 12 )
4
4
γ
1
1
ln (4π ) − + ζ ( 12 ) β( 12 )
4
4
2
√
γ
4− 2 1
1
ln (2π ) − +
ζ ( 2 ) β( 12 )
4
4
8
√
γ
4− 2 1
1
ζ ( 2 ) β( 12 ) + ln (8π ) −
4
4
4
γ
1 1
1
ζ ( ) β( 12 ) + ln (4π ) −
2 2
4
4
√
γ
4− 2 1
1
ζ ( 2 ) β( 12 ) + ln (2π ) −
8
4
4
n 2 K 0 (2π mn) =
3
16π 2
β( 32 ) ζ ( 32 ) −
2π 2
9
Hautot [73] was also able to convert the Bessel function sum (1.3.36) into a
sum of elementary functions by using the Poisson summation formula. This result,
namely
d(1) =
∞
∞
1
1
(−1)m cosech(m 2 + p 2 )1/2 π
9
π
− ln 2 + 12
,
2
2
(m 2 + p 2 )1/2
(1.3.37)
is by far the simplest formula for evaluating d(1). One requires only nine terms
to give d(1) to 10 decimal places and its evaluation on an HP35 takes about a
minute – an immense improvement on the traditional Ewald approach. However,
the derivation of (1.3.37) by Hautot is not simple and furthermore (1.3.37) is just
one of a large class of similar formulae, all of which may be derived in a fairly
1.3 The theta-function method in the analysis of lattice sums
45
elementary manner using MTθ Fs together with the Poisson summation formula.
We will apply this method to the sums a(1), b(1), c(1), and d(1), which are of
great importance in the evaluation of μ for ionic crystals (Zucker [143], [144]).
First we point out that the Poisson summation formula is essentially a transformation formula for θ3 (Bellman [6]). In fact (1.3.34) may be written
t 1/2 θ3 (e−π t ) = θ3 (e−π/t ).
(1.3.38)
In addition to this we have other Poisson-type formulae, such as
t 1/2 θ2 (e−π/t ) = θ4 (e−π/t )
(2t)1/2 θ5 (e−π t ) = θ5 (e−π/4t ).
(1.3.39)
Now, by applying these results to MTθ Fs we are able to get functional relations between a(2s) and a(2s − 3) and similar ones for other lattice sums. Let
us illustrate this for a one-dimensional sum.
Consider (m 2 )−s = ζ (2s) = Ms [θ3 − 1]. Using (1.3.26) we transform this
into Ms (θ3 − 1) = π s Ms [θ3 (q π ) − 1]. By applying the Poisson formula (1.3.38)
we have
and
Ms [θ3 − 1] = π s Ms [t −1/2 θ3 (e−π/t ) − 1].
(1.3.40)
Transforming from t to 1/t within the integral on the right-hand side of (1.3.40)
we have
Ms [θ3 − 1] =
π 2s−1 ( 12 − s)
M1/2−s [θ3 − 1],
(s)
(1.3.41)
π 2s−1 ( 12 − s)
ζ (2s) =
ζ (1 − 2s).
(s)
This is nothing other than the functional equation for the Riemann zeta function,
and the above derivation is a standard method for obtaining this relation (Titchmarsh [131]). The structure of (1.3.41) is worth noting. The function ζ (2s) has
a simple pole at s = 12 , so (1.3.41) is a reflection formula for ζ (2s) about the
point s = 14 and we can thus calculate ζ (2s) for all s, both positive and negative,
except at the pole. That a(2s) may be regarded as the three-dimensional analogue
of ζ (2s) has already been pointed out. By applying a similar technique to a(2s),
we obtain
a(2s) =
π 2s−3/2 ( 32 − s)
a(3 − 2s).
(s)
(1.3.42)
The sum a(2s) has a simple pole at s = 32 and Equation (1.3.42) is a reflection
formula for a(2s) about the point s = 34 ; we can calculate a(2s) for all s, both
46
Lattice sums
positive and negative, except at the pole. Now by using both (1.3.38) and (1.3.39)
we can find similar functional relations for b(2s), c(2s), and d(2s):
22s b(2s) = K [a(3 − 2s) + b(3 − 2s) − c(3 − 2s) − d(3 − 2s)],
22s c(2s) = K [a(3 − 2s) − b(3 − 2s) − c(3 − 2s) + d(3 − 2s)],
22s d(2s) = K [a(3 − 2s) − 3b(3 − 2s) + 3c(3 − 2s) − d(3 − 2s)], (1.3.43)
π 2s−3/2 ( 32 − s)
K =
.
(s)
If we put s =
1
2
in these results, we find the following interesting relations:
πa(1) = a(2),
π b(1) = 2b(2) + c(2),
(1.3.44)
π c(1) = b(2) + c(2) + d(2),
π d(l) = 3c(2).
Furthermore, from (1.3.30) we have
3a(1) = 3b(1)+3c(1)+d(1)
a(2) = 3b(2)+3c(2)+d(2). (1.3.45)
and
Thus, of, the eight unknowns a(1)−d(1) and a(2)−d(2), only three are independent, say b(2), c(2), and d(2), and we should like to evaluate them. In their forms
given by (1.3.29) they are hardly more rapidly convergent than b(1) − d(1), but
by means of elementary formulas they may be transformed into very rapidly converging series. This will now be illustrated by evaluating d(1) from the relation
π d(1) = 3c(2). The elementary summation formulas used are
∞
2π
π
2
= M1 [θ3 e−x t ], (1.3.46)
(m 2 + x 2 )−1 = +
2π
x
x
x(e
− 1)
−∞
∞
π cosech π x
2
= M1 [θ4 e−x t ],
(−1)m (m 2 + x 2 )−1 =
x
−∞
(1.3.47)
∞
2π
π
2
= M1 [θ2 e−x t ], (1.3.48)
[(m − 12 )2 + x 2 ]−1 = −
2π
x + 1)
x
x(e
−∞
Now, c(2) =
(−1)m+n (m 2 + n 2 + p 2 )−1 = M1 [θ42 θ3 − 1] and we write
∞
n 2 and T =
n n2
θ3 = 1 + 2S and θ4 = 1 + 2T , where S = ∞
1 q
1 (−1) q .
Then,
M1 [θ42 θ3 − 1] = M1 [θ4 (2T + 1)(2S + 1) − 1]
= M1 [4θ4 ST + θ3 θ4 + θ42 − θ4 − 1]
= M1 [θ3 θ4 − 1] + M1 [θ42 − 1] − M1 [θ4 − 1]
∞ ∞
2
2
+ 4M1 θ4
(−1)m e−t (m +n ) .
(1.3.49)
1
1
1.3 The theta-function method in the analysis of lattice sums
47
The first three parts of (1.3.49) are easily evaluated by using the results of
Glasser [56] and Zucker [58]; the last part is just (1.3.47) with x in the form of
the sum of two squares. Therefore we have
M1 [θ3 θ4 − 1] = −
M1 [θ42 − 1] = −π ln 2,
π
ln 2,
2
M1 [θ4 − 1] = −
π2
,
6
and
c(2) = −
∞
∞
1
1
3π ln 2 π 2
cosech (m 2 + n 2 )1/2 π
+
+ 4π
(−1)m
. (1.3.50)
2
6
(m 2 + n 2 )1/2
Since d(1) = 3c(2)/π , we recover Hautot’s formula. We can rearrange the
expression for c(2) as M1 [θ3 θ42 − 1], so
M1 [θ3 θ42 − 1] = M1 [θ3 (2T + 1)2 − 1]
= M1 [4θ3 T 2 + 2θ3 θ4 − θ3 − 1]
= M1 [2θ3 θ4 − 2] − M1 [θ3 − 1]
∞ ∞
2
2
+ 4M1 θ3
(−1)m+n e−t (m +n ) .
1
1
Now, as before, M1 [2θ3 θ4 − 2] = −π ln 2, M1 [θ3 − 1] = π 2 /3 and using (1.3.46)
we obtain
4M1 [θ3 T 2 ] = 4π
∞ ∞
1
+ 8π
1
(−1)m+n
(m 2 + n 2 )1/2
∞ ∞
1
1
(−1)m+n
.
(m 2 + n 2 )1/2 [exp 2π(m 2 + n 2 )1/2 − 1]
The first double sum is easily evaluated (Glasser [56]), giving 4π [ln 2 −
η( 12 ) β( 12 )], so finally we obtain
√
d(1) = 9 ln 2 − π + 12( 2 − 1) ζ ( 12 ) β( 12 )
+ 24
∞ ∞
1
1
(−1)m+n
. (1.3.51)
(m 2 + n 2 )1/2 [exp 2π(m 2 + n 2 )1/2 − 1]
This is another formula of Hautot [74] and Chaba and Pathria [34], and it requires
only four terms in the double sum to give d(1) to 10 decimal places. However,
the relations among θ -functions enable us to obtain many similar formulas without much effort. For example, θ3 θ4 = θ42 (q 2 ) so that c(2) = M1 [θ4 θ3 θ4 − 1] =
M1 [θ4 θ42 (q 2 ) − 1] and, proceeding as before, we obtain
48
Lattice sums
d(1) =
√
β
π
− √ ln [2( 2 − 1)]
2
2
∞ ∞
(−1)m+n cosech [2m 2 + 2n 2 ]1/2 π
+ 12
.
(2m 2 + 2n 2 )1/2
1
(1.3.52)
1
Further formulas are
∞ ∞
√
cosech (m 2 + 2n 2 )1/2 π
d(1) = π − 3( 2 + 1) ln 2 + 24
(m 2 + 2n 2 )1/2
1
∞
∞
cosech [2m(2n + 1)]1/2 π
π
= − − 12
(−1)m+n
2
[2m(2n + 1)]1/2
= −12π
1
1
∞ ∞
sech2
1
(1.3.53)
1
1
π
[(2m − 1)2 + (2n − 1)2 ]1/2 .
2
(1.3.54)
(1.3.55)
This last result is Mackenzie’s [102]. All these equations may be obtained by rearranging and transforming θ3 θ42 and all are formulas which lead to the evaluation
of c(2) and d(1) in an extremely rapid way. This profusion of results is rather
embarrassing as there appears to be no rhyme or reason in them. Without doubt
there are many others of a similar nature and, in view of the later calculations in
Section 1.5, this raises the question whether any of these formulae will lead to a
closed form for d(1) in known transcendental constants. Certainly all the diagonal
terms in the double sums in (1.3.50)–(1.3.55) may be evaluated. For example, if
we put n = m in (1.3.52) we have
12π
∞
cosech 2mπ
1
2m
=π−
9
ln 2,
2
(1.3.56)
and this is the
√ major contribution in the double sum. In fact we are able to sum
∞
1 (cosech λnπ )/n in terms of π and logarithms of surds for any rational λ,
so that a closed-form result for d(1) might yet be obtained.
In a similar way expressions for b(2) and d(2) may be obtained and hence also
for b(1) and c(1). Again, results for e(2) and h(2) may be found and these are
related to e(1) and h(1) by the equations
√
π e(1) = f (2),
π f (1) = e(2),
πg(1) = 2g(2),
(1.3.57)
π h(1) = h(2) + j (2),
π j (1) = h(2) − j (2).
In Table 1.4 we give some rapidly convergent double sums which can be used to
evaluate b(2) and h(2). We emphasize again that these are not unique and many
others may be found using θ -function relations. Nor are these methods restricted
to s = 12 and 1. The intriguing formula
3b(3) + 3c(3) + d(3) = −4π ln 2
(1.3.58)
1.3 The theta-function method in the analysis of lattice sums
49
Table 1.4 Exponential and hyperbolic forms for the basic lattice sums with s = 1
b(2) =
√ √
√
1
4π 2 √
1
23 + 17 2 1/2
β
− 2π −
2π 3 ln 2 − 4η
− 2( 2 + 1)
− 2π ln
2
2
3
2
∞
∞ √ + 8 2π
1
1
(−1)m+n
√
(m 2 + n 2 )1/2 [exp 4 2π(m 2 + n 2 )1/2 − 1]
∞
∞ √
π
(−1)m+n cosech (2m 2 + 2n 2 )1/2 π
π2
− √ ln 2
2 + 1 + 4π
c(2) =
6
(2m 2 + 2n 2 )1/2
2
1
1
d(2) =
e(2) =
∞
∞ √
cosech π(m 2 + 2n 2 )1/2
π2
π
2 + 1 + 8π
(−1)n
− π ln 2 − √ ln 2
3
(m 2 + 2n 2 )1/2
2
1
1
√
√
2π ln (1 + 2) + 8π
∞
∞ 1
1
1 2
cosech π 2m 2 + 2 n −
2
2 1/2
1
2m 2 + 2 n −
2
1/2
2
2+ n− 1
cosech
π
m
∞
∞ 2
√
√
f (2) = 2π ln (1 + 2) + 4π
(−1)m
2 1/2
1
1
1
m2 + n −
2
g(2) = 2π
∞
∞ cosech π
(−1)n
1
2
2n −
1
2
2
+ (m −
1 2
)
2
1/2
1/2
1/2
1 2
1 2
2n −
+ m−
2
2
⎧
⎪
1 2
⎪
⎪
⎪
cosech π 2m 2 + n −
⎪
∞
∞
⎨
2
√
√
h(2) = π(4 − 2) ln (1 + 2) + 8π
2 1/2
⎪
⎪
1
1
1 ⎪
⎪
⎪
2m 2 + n −
⎩
2
−∞ −∞
1
2
1/2
1 2
(−1)m cosech π 2m 2 + 2 n −
2
−
2 1/2
1
2m 2 + 2 n −
2
j (2) =
√
√
2π ln (1 + 2) + 8π
∞
∞ 1
1
1/2
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
1/2
1 2
cosech π 2m 2 + 2 n −
(−1)m
2
1/2
1 2
2m 2 + 2 n −
2
and results for larger-s sums may be obtained from higher-order analogues of
(1.3.46)–(1.3.48). Thus, by differentiating (1.3.46) with respect to x we have
∞
−∞
(m 2 + x 2 )−2 =
π cosech2 π x
π
π
+
+
,
2x 3
x 3 (e2π x − 1)
2x 2
(1.3.59)
50
Lattice sums
and so, for example, to evaluate a(4) = M2 [θ33 − 1] we find, following the
procedure previously described, that
(m 2 + n 2 + p 2 )−2
a(4) =
= 8ζ (2) β(2) − 2ζ (4) + 2π [ζ ( 32 ) β( 32 ) − ζ (3)]
+ 4π
∞ ∞
+ 4π 2
1
(m 2
1
1
∞
∞ 1
1
+ n 2 )3/2 [exp 2π(m 2
+ n 2 )1/2 − 1]
cosech2 π(m 2 + n 2 )1/2
.
2(m 2 + n 2 )
(1.3.60)
The sum a(4) is one of the most slowly convergent sums, yet the above formula
gives 10 significant figures if one retains only the first three terms from each
double sum. A sequence of such formulas enables us to evaluate a(2s) and j (2s).
These lattice sums are given in Table 1.5. We have dealt here only with cubic
lattice sums, but the MTθ F method can be used for tetragonal and orthorhombic
crystals (Zucker [143]), and hexagonal crystals can undoubtedly be handled this
way though no analysis has yet been attempted.
We conclude this section with a short account of some recent related work by
Chaba and Pathria [32–35]. Though Chaba and Pathria do not explicitly make use
of MTθ Fs, they use the Poisson summation formula to obtain many results similar
to those described above. They consider n-dimensional sums of the form
exp (−a R 2 )/R 2s ,
(a)
cos (2π ε1l1 ) · · · cos (2π εn ln ) exp (−a R 2 )/R 2s , (1.3.61)
(b)
exp [−a(l1 + ε1 )2 ] · · · exp [−a(ln + εn )2 ]
(c)
× [(l1 + ε1 )2 + · · · + (ln + εn )2 ]−s ,
where R 2 = l12 + l22 + · · · + ln2 . (A two-dimensional sum of this type was originally considered by Glasser [58].) Chaba and Pathria [32] then find a number of
transformation formulas, e.g.,
exp (−a R 2 )
π 2 R2
π n/2
+ (n−2)/2 E (4−n)/2
a
R2
a
2
=
π n/2 a (2−n)/2 + a + Cn (n = 1, 3, 4),
n−2
= −π ln a + a + C2 (n = 2),
(1.3.62)
where
∞
Em =
1
t −m exp (−zt) dt,
m > 0.
(1.3.63)
1.3 The theta-function method in the analysis of lattice sums
Table 1.5 Numerical values of the basic lattice sumsa
s
a(2s)
b(2s)
c(2s)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
−2.8372974794(0)
−8.913632917(0)
∞
1.653231596(1)
1.037752483(1)
8.401920546(0)
7.467057780(0)
6.945807926(0)
6.628859200(0)
6.426119101(0)
6.292294498(0)
6.202149043(0)
6.140599581(0)
6.098184126(0)
6.068764295(0)
6.048263469(0)
−7.7438614142(−1)
−3.013802247(−1)
2.2228710876(−1)
6.8922257409(−1)
1.0619823625(0)
1.3411086694(0)
1.5422045834(0)
1.6837514910(0)
1.7820445621(0)
1.8498105369(0)
1.8963845503(0)
1.9283773751(0)
1.9503780885(0)
1.9655380752(0)
1.9760103762(0)
1.9832637191(0)
−1.4803898065(0)
−1.8300453641(0)
−2.0461936439(0)
−2.1568872986(0)
−2.1968298536(0)
−2.1955219241(0)
−2.1738503493(0)
−2.1448133764(0)
−2.1155844408(0)
−2.0895601364(0)
−2.0679279901(0)
−2.0507132697(0)
−2.0374137084(0)
−2.0273527819(0)
−2.0198580752(0)
−2.0143388512(0)
s
d(2s)
e(2s)
f (2s)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
−1.7475645946(0)
−2.5193561521(0)
−3.2386247661(0)
−3.8631638072(0)
−4.3786011499(0)
−4.7884437139(0)
−5.1054293720(0)
−5.3455657733(0)
−5.5246636380(0)
−5.6566662484(0)
−5.7530849044(0)
−5.8230277889(0)
−5.8734960409(0)
−5.9097623766(0)
−5.9357395661(0)
−5.9542997599(0)
1.5401709012(0)
4.1458674219(0)
7.893704872(0)
1.3054752185(1)
2.0132691063(1)
2.9906494892(1)
4.3507844901(1)
6.2547665025(1)
8.9305752590(1)
1.2700438916(2)
1.8019768497(2)
2.5532313378(2)
3.6148194853(2)
5.1154270695(2)
7.2370214795(2)
1.0236924529(3)
1.3196705870(0)
4.8385895905(0)
1.2568251129(1)
2.8477747226(1)
6.0522064903(1)
1.2465784291(2)
2.5284972479(2)
5.0907147682(2)
1.0213048420(3)
2.0455377585(3)
4.0937626666(3)
8.1899751417(3)
1.6382172875(4)
3.2766354948(4)
6.5534521329(4)
1.3107067253(5)
s
g(2s)
h(2s)
j (2s)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
2.5335574044(0)
5.6281494825(0)
8.7850400797(0)
1.1857062518(1)
1.4885926393(1)
1.7985164284(1)
2.1283071340(1)
2.4902898412(1)
2.8960408839(1)
3.3567795759(1)
3.8839124460(1)
4.4895602845(1)
5.1870317651(1)
5.9912550183(1)
6.9191909161(1)
7.2368819704(2)
3.634989014(0)
7.7294890464(0)
1.2163847976(1)
1.7414440838(1)
2.4235087473(1)
3.3588428415(1)
4.6717592062(1)
6.5293708145(1)
9.1625607686(1)
1.2894706095(2)
1.8181430446(2)
2.5666225287(2)
3.6258741613(2)
5.1245294757(2)
7.2445017880(2)
1.0243062668(3)
1.2858465498(0)
3.6898829776(0)
7.3479876218(0)
1.2517226760(1)
1.9662159451(1)
2.9525897876(1)
4.3216738240(1)
6.2333952840(1)
8.9153618774(1)
1.2689861363(2)
1.8012547440(2)
2.5527454010(2)
3.6144961780(2)
5.1152139126(2)
7.2368819704(2)
1.0236833762(3)
a The values given must each be multiplied by 10n , where n is the number in
parentheses.
51
52
Lattice sums
By putting a = π , several interesting results may be obtained. It is shown that
C1 = π 2 /3, that C2 = π(γ − η), where γ is Euler’s constant, and that η =
ln [( 14 )4 ]/4π 3 . The constant C3 turns out to be equal to a(2), and C4 was shown
by one of us to be −8 ln 2. This gives the attractive formula
exp [−π(l 2 + l 2 + l 2 + l 2 )]
1
2
3
4
l12 + l22 + l32 + l42
= π − 4 ln 2.
(1.3.64)
A sequence of such formulas was obtained, of which the most interesting are
some exact results for double sums involving exponential functions similar to that
which appears in (1.3.51). For example,
∞ ∞
1
1
+ n 2 )1/2 + (−1)m ]
√
π
2− 2 1
9
η
ln 2 −
+
ζ ( 2 ) β( 12 ) − ,
=
16
16
2
8
1 4
( 4 )
η = ln
4π 3
1
(m 2
+ n 2 )1/2 [exp (m 2
(1.3.65)
(Chaba and Pathria [34]). Another exact result was derived by one of us from
consideration of the dipole sum treated by Nijboer and de Wette [111]. This is
∞ ∞
1
1
1
(m 2 + n 2 )1/2
=
24
exp [2π(m 2 + n 2 )1/2 ] − 1
ζ ( 3 ) β( 3 )
1
−1 + 2 2 2 .
π
8π
(1.3.66)
Formulae such as (1.3.65) and (1.3.66) give us hope that many three-dimensional
sums may yet be summed exactly. Chaba and Pathria [35] also found several results similar to (1.3.35). In addition, they obtained some one-dimensional
formulae similar to (1.3.65) and (1.3.66), such as
∞
m −1 (e2π m − 1)−1 =
1
π
1
1
ln 2 −
− η,
2
12 4
(1.3.67)
but these results can, in fact, be generalized. In the notation of Whittaker and Watson [139], let K (k) be the complete elliptic integral of the first kind of modulus k:
K =
0
1
dx
.
[(1 − x 2 )(1 − k 2 x 2 )]1/2
With k 2 + k 2 = 1 and K (k ) = K , if (K /K )2 = λ where λ, is rational, it is
shown in Appendix 1.7 at the end of the chapter that our exact results for certain
lattice sums permit us to express K in terms of gamma functions. By using these
results the following sums can be evaluated for all rational λ (Zucker [146]):
1.3 The theta-function method in the analysis of lattice sums
∞
m
−1
1
53
√
√
1 2K 3 kk π λ
−1
[exp (2π m λ) − 1] = − ln
−
6
12
π3
√
∞
√
k2
1
π λ
m −1
−1
(−1) m [exp (2π m λ) − 1] = − ln
−
12 16k 12
1
∞
m
1
−1
√
√
π λ
1 kK
−1
+
[exp (2π m λ) + 1] = ln
2 2π
4
√ (1.3.68)
∞
√
2k
1
π λ
m −1
−1
(−1) m [exp (2π m λ) + 1] = ln
−
2 1 − k
4
1
∞
1
√
1 2k K
(2m − 1)−1 {exp [(2 − 1)π λ] − 1}−1 = − ln
4
π
∞
√
1 2K
.
(2m − 1)−1 {exp [(2 − 1)π λ] + 1}−1 = ln
4
π
1
Along with such sums, with each crystal lattice there is associated a class of
multiple integrals related to the Green’s functions for a variety of physical processes. A subclass of these integrals, now called Watson integrals, described in
Appendix 1.8, has been shown to be expressible in terms of K values for which
λ is rational. Consequently there appears to be an intriguing connection between
lattice sums and Watson integrals, whose elucidation could well have practical
significance.
In concluding this section, we should like to mention some recent work by Hall
[68–70] and Barber [5] concerning the asymptotic behaviour of the lattice sums
(1.3.61a). The asymptotic properties of these as a function of the parameter a have
been studied. Hall found for two particular types of lattice sum that the asymptotic
behaviour was analogous to that found in the theory of critical phenomena. In
particular, the asymptotic behaviour was often independent of lattice structure,
thus exhibiting a kind of universality. Barber [5] investigated sums of the form
S(k, β) =
R−s f (k · R) e−a R
2
in the limit β → 0, |k| → 0 with β/|k| ∼ 1 and showed that these sums
also exhibit crossover phenomena. The possible connection of lattice sums with
critical phenomena should be worth exploring.
The class of sums
Sp =
m,n
(−1)m+n+mn
exp [−(π/2)(m 2 + n 2 )]
(m + in)2 p
54
Lattice sums
was studied by Boon and Zak [16]. These arise out of the completeness relation for
coherent oscillator states on a von Neumann lattice and appear to be related to the
invariants of Weierstrass elliptic functions. Some of their interesting properties are
S1 = S3 = 0,
S2 = − 14
S4 =
19 2
S ,
14 2
(m + in)−4 = −
m,n
8 ( 14 )
.
1024π 2
1.4 Number-theoretic approaches to lattice sums
We pointed out earlier that Jones and Ingham [85] were the first to introduce
number-theoretic considerations into the lattice sum problem. This idea was subsequently elaborated by Emersleben [46]. The principal aim, however, was to use
number-theoretic tables, such as the representation functions for an integer as a
sum of squares, as an aid in the numerical evaluation of sums. In this section we
shall describe the application of number-theoretic ideas to the exact evaluation of
a variety of multiple sums. In accordance with the exact results obtained already,
it is reasonable to say that a lattice sum is solvable when it has been expressed in
terms of a finite number of one-dimensional sums such as the Riemann zeta function. These are the Dirichlet L-series, and we shall begin with a brief survey of
their more important properties. An elementary L-series (modulo k) is a function
of the form
∞
χk (n) n −s ,
(1.4.1)
L k (s, χ ) =
n=1
where χk (a Dirichlet character) takes on the values ±1, 0 and has the properties
χk (1) = 1,
χk (n + k) = χk (n),
χk (n) = 0
χk (nm) = χk (m) χk (n),
if k and n have a common factor d = ±1.
In particular,
ζ (s) = L 1 (s, 1).
(1.4.2)
The most important properties of these functions were collected and proved by
Zucker and Robertson [147]. For example, let P denote a square-free odd integer
divisible by the prime p. Then
(1) if k = 2P, P p , or 2α P (α > 3), there are no elementary L-series;
(2) if k = P, 4P, there is just one such L-series;
(3) if k = 8P, there are exactly two such series.
1.4 Number-theoretic approaches to lattice sums
55
Thus, L 3 = 1 − 2−s + 4−s − 5−s + · · · . One might suspect that the corresponding
series with constant signs is an L-series for k = 3, but it is easily seen that this
series is (1 − 3−s )L 1 . The two elementary series for k = 24 are
L 24a = 1 + 5−s + 7−s + 11−s − 13−s − 17−s − 19−s − 23−s + · · · ,
L 24b = 1 + 5−s − 7−s − 11−s − 13−s − 17−s + 19−s + 23−s + · · · . (1.4.3)
Since for the first series χk (k − 1) = −1, we denote it L −24 ; for a similar reason,
the second series is denoted L 24 .
Each of these L-series obeys a simple functional equation,
sin (sπ/2)
L ±k (s) = C(s)
L ±k (1 − s), C(s) = 2s π s−1 k −s+1/2 (1 − s),
cos (sπ/2)
(1.4.4)
and can be summed at integer values:
L ±k (1 − 2m) =
L ±k (−2m) =
(−1)m R(2m − 1)!/(2k)2m−1 ,
0,
0,
(−1)m R (2m)!/(2k)2m ,
L k (2m) = Rk −1/2 π 2m ,
(1.4.5)
L −k (2m − 1) = R k −1/2 π 2m−1 ,
where m is a positive integer and R, R are rational numbers which depend
on m, k.
These Dirichlet series are intimately related to two-dimensional lattice sums of
the form
∞
[Q(m, n)]−s = S(a, b, c; s),
(1.4.6)
m,n=−∞
where Q(m, n) = am 2 +bmn+cn 2 is an integer quadratic form. The discriminant
of Q is the quantity d = b2 − 4ac, which we shall assume is negative so that Q
never changes sign. A unimodular transformation is a change of variables m =
αx + βy, n = γ x + δy, where α, β, γ , and δ are integers and αδ − βγ = 1. Under
this substitution Q becomes a new form Q (x, y), which is said to be equivalent
to Q. It is easily verified that equivalent forms have the same discriminant, but it is
also true that for each d < 0 there is a finite number h(d) of equivalence classes,
called the class number of d. It is a remarkable result, proved by Dirichlet, that
L −d (1) = 2π h(d)/w|d|1/2 , where
⎧
⎪
d = −3a 2 ,
⎪ 6,
⎨
(1.4.7)
w=
4,
d = −4a 2 ,
⎪
⎪
⎩
2
otherwise.
56
Lattice sums
(The number w is the number of substitutions which leave the quadratic form
unchanged.) If we know the number of different solutions, R Q (N ), to the equation
Q(m, n) = N then we can write (we assume that Q > 0)
S(a, b, c; s) =
∞
R Q (N ) N −s .
(1.4.8)
N =1
It turns out in many cases, including a number not readily solvable through
θ -function identities, that (1.4.8) can be decomposed into elementary Dirichlet
series. We first require some additional number-theoretic tools.
Two integers a, b leaving the same remainder when divided by n are said to be
congruent modulo n: a ≡ b (mod n). A number congruent to a square is called
a quadratic residue modulo n. Let p be a prime. Then the Legendre symbol is
defined by
⎧
⎪
0,
p divides m,
⎪
⎨
(1.4.9)
(m| p) =
1,
m is a quadratic residue (mod p),
⎪
⎪
⎩
−1
otherwise.
The generalization of this to composite moduli n = p1e1 · · · plel is called the
Jacobi–Kronecker symbol and is defined by
(m|n) = (m| p1 )e1 · · · (m| pl )el .
In terms of this symbol we have L ±k (s) =
Dirichlet’s formula
Rd (k) = w(d)
∞
1
(1.4.10)
(±k|n) n −s . Our principal tool is
(d|m)
(1.4.11)
m|k
for the number of ways in which an integer k is represented by a set of representations, one from each equivalence class, of the quadratic forms with discriminant
d. Equation (1.4.11) is valid when k has no divisor in common with d (usually
expressed as (k, d) = 1) and the sum is over all the divisors of k. The solvability
of S(a, b, c; s) devolves upon the ability to extract from Rd (k) the representation
function R Q (k) itself, and there is a classification of forms which suggests when
this is possible.
If there is only one class of forms with discriminant d, in which case it is
convenient to consider the reduced form, for which b ≤ a ≤ c, (1.4.11) gives
R Q (k) directly for those k which are relatively prime to d ((k, d) = 1). For other
values of k, R Q (k) can be obtained ad hoc. We shall illustrate this by the lattice
sums for square and triangular lattices:
1.4 Number-theoretic approaches to lattice sums
S(1, 0, 1; s) =
S(1, 1, 1; s) =
(m 2 + n 2 )−s ,
Q1 = m 2 + n2,
57
d = −4,
(1.4.12)
2 −s
(m + mn + n ) ,
2
Q 2 = m + mn + n ,
2
2
d = −3.
When k is odd it can have no factor in common with d = −4 so, by Dirichlet’s
formula,
(−4|m).
R Q 1 (k) = 4
m|k
If k = 2q then if 2q = x 2 + y 2 we can write x + y = 2a, x − y = 2b, so
q = a 2 + b2 is also of this form, that is,
(−1|m).
R Q 1 (2l k) = R Q 1 (k) = 4
m|k
odd
Hence
∞
∞
∞
R Q 1 (n)
R Q 1 (k)
1 R Q 1 (k)
S(1, 0, 1; s) =
=
+ s
+ ···
ns
ks
2 k=1
ks
k=1
m=1
odd
= (1 − 2−s )−1
∞
odd
R Q 1 (k) k −s
(1.4.13)
k=1
odd
= 4(1 − 2−s )
∞ k=1
odd
(−4|m) k −s .
m|k
However, we can write k = mr and sum freely over the odd values of m and r .
Therefore,
S(1, 0, 1; s) = 4(1 − 2−s )−1
∞
(−4|m) m −s
m=1
odd
= 4L −4 (s) L 1 (s) = 4 β(s) ζ (s),
∞
r −s
r =1
odd
(1.4.14)
since 1 + 1/3s + 1/5s + · · · = ζ (s) − (1/2s + 1/4s + · · · ) = ζ (s)(1 − 2−s ) and
L −4 (s) = 1 − 3−s + 5−s − 7−s + · · · ≡ β(s).
Similarly, when (k, 3) = 1,
(−3|m).
(1.4.15)
R Q 2 (k) = 6
m|k
58
Lattice sums
When 3k = m 2 + mn + n 2 , then 3 divides 3k + 3(mn + n 2 ) = (m + 2n)2 . Thus
m + 2n = 3a and m − n = 3a − 3n = 3b. Consequently, 3k = 3(a 2 + ab + b2 )
so k is also represented by Q 2 . This means that R Q 2 (3l k) = R Q 2 (k). Following
the argument above, we have
S(1, 1, 1; s) = 6 L 1 (s) L −3 (s) = 6 ζ (s) g(s),
(1.4.16)
where g(s) = 1 − 2−s + 4−s − 5−s + 7−s . . .
More generally it can happen that the reduced forms in the various h(d) equivalence classes represent disjoint sets of integers, which in number-theoretic terms
is expressed by saying that there is one class per genus. In this case there exist
functions taking on different values on the sets of integers, with whose help it
is possible to disentangle the R Q (n) from the Rd (n) as we shall illustrate with
the sum S(1, 0, 16; s). The discriminant in this case is −64, for which the class
number is 2, and so we must deal with the two reduced forms
Q 1 = m 2 + 16n 2 ,
Q 2 = 4m 2 + 4mn + 5n 2 .
(A convenient tabulation of these facts is given in L. E. Dickson [39].) An exam√
ination of the character table for the Abelian group of order 8 = −d yields the
function
⎧
⎪
0,
n even,
⎪
⎨
(1.4.17)
C(n) =
1,
n = ±1 (mod 8),
⎪
⎪
⎩
−1,
n = ±3 (mod 8).
It is easily verified that this has the property
C(k) =
1,
k = Q 1 (m, n) odd,
−1,
k = Q 2 (m, n) odd.
Hence when k is odd we find from Dirichlet’s formula that
R Q 1 (k) = [C(k) + 1]
(−64|m).
(1.4.18)
(1.4.19)
m|k
When 2k = x 2 + 16y 2 , k is of the form 2(a 2 + 4y 2 ), which is impossible. Hence
R Q 1 (2k) = 0. If 4k = x 2 + 16y 2 then k must have the form a 2 + 4y 2 . The representation function for this quadratic form can be worked out as in the preceding
paragraph, and we obtain
(−16|m).
R Q 1 (4k) = 2
m|k
1.4 Number-theoretic approaches to lattice sums
59
When 8k = x 2 + 16y 2 , k has the form 2(a 2 + y 2 ) so again R Q 1 (8k) = 0. Finally,
when 2l k = x 2 + 16y 2 , l ≥ 4, factoring out factors of 2 leads again to the
two-squares problem and
R Q 1 (2l k) = 4
(−4|m).
m|k
This specifies R Q 1 (n) completely and the summation procedure used above
yields
S(1, 0, 16; s) = (1 − 2−s + 21−2s − 21−3s + 22−4s )L 1 (s) L −4 (s)
+L 8 (s) L −8 (s).
(1.4.20)
This procedure was applied systematically by Zucker and Robertson [148] and
leads to the results in Table 1.6.
When there is more than one class per genus, that is, when the reduced forms
corresponding to the discriminant d represent integers in common, the situation is
exceedingly complex. Indeed, we conjecture (Glasser [57]) that only lattice sums
which correspond to discriminants having one class per genus are solvable. This
has been verified for discriminants |d| 40000 and so far no lattice sum outside
this class has been solved. The proof of this conjecture remains an interesting
open question.
In addition to S(a, b, c; s) it is also desirable to consider the alternating sums
(−1)m Q −s ,
S1 (a, b, c; s) =
S2 (a, b, c; s) =
S1,2 (a, b, c; s) =
(−1)n Q −s ,
(1.4.21)
(−1)m+n Q −s .
There appears to be no systematic approach to these, but they are not independent
of the basic sum S and the following relations have been derived (the details can
be found in Zucker and Robertson [147, 148]):
S(1, 0, λ; s) + S1 (1, 0, λ; s) + S2 (1, 0, λ; s) + S1,2 (1, 0, λ; s)
= 22−2s S(1, 0, λ; s),
S(1, 0, λ; s) + S2 (1, 0, λ; s) = 2 S(1, 0, 4λ; s),
S(1, 1, λ ; s) + S2 (1, 1, λ ; s) = 2 S(1, 0, λ; s)
(λ = 14 (1 + λ)),
S1,2 (1, 1, λ ; s) = S1 (1, 1, λ ; s) = S1,2 (1, 0, λ; s),
S(1, 1, λ ; s) + S1 (1, 1, λ ; s) + S2 (1, 1, λ ; s) + S1,2 (1, 1, λ ; s)
= 22−2s S(1, 1, λ ; s).
(1.4.22)
60
Lattice sums
Table 1.6 Principal solutions of all solvable S(a, b, c; s) for various c-values and
(i) a = 1, b = 0, (ii) a = 1, b = 1
c
(i) a = 1, b = 0
1
2
3
4
5
6
7
8
9
10
12
13
15
16
18
21
22
24
25
28
30
33
37
40
42
45
4L +1 L −4
2L +1 L −8
2(1 + 21−2s )L +1 L −3
2(1 − 2−s + 21−2s )L +1 L −4
L +1 L −20 + L −4 L +5
L +1 L −24 + L −3 L +8
2(1 − 21−s + 21−2s )L +1 L −7
(1 − 2−s + 21−2s )L +1 L −8 + L −4 L +8
(1 + 31−2s )L +1 L −4 + L −3 L +12
L +1 L −40 + L +5 L −8
(1 − 2−2s + 22−4s )L +1 L −3 + L −4 L +12
L +1 L −52 + L −4 L +13
(1 − 21−s + 21−2s )L +1 L −15 + (1 + 21−s + 21−2s )L −3 L +5
(1 − 2−s + 21−2s − 21−3s + 22−4s )L +1 L −4 + L −8 L +8
(1 − 2 × 3−s + 31−2s )L +1 L −8 + L −3 L +24
2−1 (L +1 L −84 + L −3 L +28 + L −7 L +12 + L −4 L +21 )
L +1 L −88 + L −11 L +8
2−1 [(1 − 2−s + 21−2s )L +1 L −24 + (1 + 2−s + 21−2s )L −3 L +8 + L −4 L +24 + L −8 L +12 ]
(1 − 2.5−s + 51−2s )L +1 L −4 + L +5 L −20
(1 − 21−s + 3 × 2−2s − 22−3s + 22−4s )L +1 L −7 + L −4 L +28
2−1 [L +1 L −120 + L −3 L +40 + L +5 L −24 + L +8 L −15 ]
2−1 [L +1 L −132 + L −3 L +44 + L −11 L +12 + L −4 L +33 ]
L +1 L −148 + L −4 L +37
2−1 [(1 − 2−s + 21−2s )L +1 L −40 + (1 + 2−s + 21−2s )L +5 L −8 + L −4 L +40 + L −8 L +20 ]
2−1 [L +1 L −168 + L −3 L +56 + L −7 L +24 + L −8 L +21 ]
2−1 [(1 − 2 × 3−s + 31−2s )L +1 L −20 + (1 + 2 × 3−s + 31−2s )L −4 L +15 + L −3 L +60
+ L +12 L −15 ]
2−1 [(1 + 2−2s + 21−4s + 23−6s )L +1 L −3 + (1 + 21−2s )L −4 L +12 + L −8 L +24
+ L +8 L −24 ]
2−1 [L +1 L −228 + L −3 L +76 + L −19 L +12 + L −4 L +57 ]
L +1 L −232 + L −8 L +29
2−1 [(1 − 21−s + 3 × 2−2s − 22−3s + 22−4s )L +1 L −15 + L −4 L +60
+ (1 + 21−s + 3 × 2−2s + 22−3s + 22−4s )L −3 L +5 + L +12 L −20 ]
2−1 [L +1 L −180 + L +5 L −56 + L −7 L +40 + L +8 L −35 ]
2−1 [(1 − 2.3−s + 31−2s )(1 − 2−s + 21−2s )L +1 L −8 + L +12 L −24
+ (1 + 2 × 3−s + 31−2s )L −4 L +8 + (1 + 2−s + 21−2s )L −3 L +24 ]
2−1 [L +1 L −312 + L −3 L +104 + L +13 L −24 + L +8 L −39 ]
2−1 [L +1 L −340 + L +5 L −68 + L −20 + L +17 + L −4 L +85 ]
2−1 [(1 − 2−s + 21−2s )L +1 L −88 + (1 + 2−s + 21−2s )L −11 L +8 + L −4 L +88 + L −8 L +44 ]
2−1 [L +1 L −372 + L −3 L +124 + L −31 L +12 + L −4 L +93 ]
2−1 [L +1 L −408 + L −3 L +136 + L +17 L −24 + L +8 L −51 ]
2−2 [L +1 L −420 + L −3 L +140 + L +5 L −84 + L −7 L +60
+ L −15 L +28 + L +21 L −20 + L −35 L +12 + L −4 L +105 ]
2−1 [(1 − 21−s + 3 × 2−2s − 22−3s + 3 × 21−4s − 23−5s + 23−6s )L +1 L −7
+ (1 + 21−2s )L −4 L +28 + L −8 L +56 + L +8 L −56 ]
2−2 [(1 − 2−s + 21−2s )L +1 L −120 + L −4 L +120 + (1 + 2−s + 21−2s )L −3 L +40
+ L +12 L −40 + (1 + 2−s + 21−2s )L +5 L −24 + L −20 L +24 + (1 − 2−s + 21−2s )L −15
L +8 + L −8 L +60
2−1 [L +1 L −520 + L +5 L −104 + L +13 L −40 + L −8 L +65 ]
2−1 [L +1 L −532 + L −7 L +76 + L −19 L +28 + L −4 L +133 ]
2−2 [L +1 L −660 + L +5 L −132 + L −3 L +220 + L −11 L +60
+ L −15 L +44 + L +33 L −20 + L −55 L +12 + L −4 L +165 ]
48
57
58
60
70
72
78
85
88
93
102
105
112
120
130
133
165
1.4 Number-theoretic approaches to lattice sums
61
Table 1.6 (cont.)
c
(i) a = 1, b = 0
168
2−2 [(1 − 2−s + 21−2s )L +1 L −168 + L −4 L +168 + (1 + 2−s + 21−2s )L −3 L +56
+ L +12 L −56 + (1 − 2−s + 21−2s )L −7 L +24 + L +28 L −24 + (1 + 2−s + 21−2s )
× L −8 L +21 + L +8 L −84 ]
2−1 [L +1 L −708 + L −3 L +236 + L −59 L +12 + L −4 L +177 ]
2−1 [L +1 L −760 + L +5 L −152 + L −19 L +40 + L +8 L −95 ]
2−2 [L +1 L −840 + L −3 L +280 + L +5 L −168 + L −7 L +120
+ L −15 L +56 + L +21 L −40 + L −35 L +24 + L −8 L +105 ]
2−1 [(1 − 2−s + 21−2s )L +1 L −232 + L −4 L +232 + (1 + 2−s + 21−2s )L −8 L +29
+ L +8 L −116 ]
2−2 [(1 − 21−s + 3 × 2−2s − 22−3s + 3 × 21−4s − 22−5s + 22−6s )L +1 L −15 +
(1+21−s + 3 × 2−2s + 22−3s + 3 × 21−4s + 22−5s + 22−6s )L −3 L +5 + (1 + 21−2s )
× L −4 L +60 + (1 + 21−2s )L +12 L −20 + L −24 L +40 + L +24 L −40 + L −8 L +120
+ L +8 L −120 ]
2−1 [L +1 L −1012 + L −11 L +92 + L −23 L +44 + L −4 L +253 ]
2−2 [L +1 L −1092 + L −3 L +364 + L −7 L +156 + L +13 L −84
+ L +21 L −52 + L −39 L +28 + L −91 L +12 + L −4 L +273 ]
2−2 [(1 − 2−s + 21−2s )L +1 L −280 + L −4 L +280 + (1 + 2−s + 21−2s )L +5 L −56
+ L −20 L +56 + (1 − 2−s + 21−2s )L −7 L +40 + L +28 L −40 + (1 + 2−s + 21−2s )
× L −35 L +8 + L −8 L +140 ]
2−2 [(1 − 2−s + 21−2s )L +1 L −312 + L −4 L +312 + (1 + 2−s + 21−2s )L −3 L +104
+ L +12 L −104 + (1 + 2−s + 21−2s )L +13 L −24 + L −52 L +24 + (1 − 2−s + 21−2s )
× L −39 L +8 + L −8 L +156 ]
2−2 [L +1 L −1320 + L −3 L +440 + L +5 L −264 + L −11 L +120
+ L −15 L +88 + L +33 L −40 + L −55 L +24 + L −8 L +165 ]
2−2 [L +1 L −1380 + L −3 L +460 + L +5 L −276 + L −23 L +60
+ L −15 L +92 + L +69 L −20 + L −115 L +12 + L −4 L +345 ]
2−2 [L +1 L −1428 + L −3 L +476 + L −7 L +204 + L +17 L −84
+ L +21 L −68 + L −51 L +28 + L −119 L +12 + L −4 L +357 ]
2−2 [L +1 L −1540 + L +5 L −308 + L −7 L +220 + L −11 L +140
+ L −55 L +28 + L −35 L +44 + L +77 L −20 + L −4 L +385 ]
2−2 [(1 − 2−s + 21−2s )L +1 L −408 + L −4 L +408 + (1 + 2−s + 21−2s )L −3 L +136
+ L +12 L −136 + (1 − 2−s + 21−2s )L +17 L −24 + L −68 L +24 + (1 + 2−s + 21−2s )
× L −51 L +8 + L −8 L +204 ]
2−2 [L +1 L −1848 + L −3 L +616 + L −7 L +264 + L −11 L +168
+ L +21 L −88 + L +33 L −56 + L +77 L −24 + L +8 L −231 ]
2−2 [(1 − 2−s + 21−2s )L +1 L −520 + L −4 L +520 + (1 + 2−s + 21−2s )L +5 L −104
+ L −20 L +104 + (1 + 2−s + 21−2s )L +13 L −40 + L −52 L +40 + (1 − 2−s + 21−2s )
× L −8 L +65 + L +8 L −260 ]
2−2 [(1 − 2−s + 21−2s )L +1 L −760 + L −4 L +760 + (1 + 2−s + 21−2s )L +5 L −162
+ L −20 L +162 + (1 + 2−s + 21−2s )L −19 L +40 + L +76 L −40 + (1 − 2−s + 21−2s )
× L −95 L +8 + L −8 L +380 ]
2−3 [(1 − 2−s + 21−2s )L +1 L −840 + L −4 L +840 + (1 + 2−s + 21−2s )L −3 L +280
+ L +12 L −280 + (1 + 2−s + 21−2s )L +5 L −168 + L −20 L +168 + (1 − 2−s + 21−2s )
× L −7 L +120 + L +28 L −120 + (1 − 2−s + 21−2s )L −15 L +56 + L +60 L −56 + (1 + 2−s
+ 21−2s )L +21 L −40 + L −84 L +40 + (1 + 2−s + 21−2s )L −35 L +24 + L +140 L −24
+ (1 − 2−s + 21−2s )L +105 L +8 + L −420 L +8 ]
−3
2 [(1 − 2−s + 21−2s )L +1 L −1320 + L −4 L +1320 + (1 + 2−s + 21−2s )L −3 L +440
+ L +12 L −440 + (1 + 2−s + 21−2s )L +5 L −264 + L −20 L +264 + (1 + 2−s + 21−2s )
× L −11 L +120 + L +44 L −120 + (1 − 2−s + 21−2s )L −15 L +88 + L +60 L −88
+ (1 − 2−s + 21−2s )L +33 L −40 + L −132 L +40 + (1 − 2−s + 21−2s )L −55 L +24
+ L +220 L −24 + (1 + 2−s + 21−2s )L +165 L −8 + L +8 L −660 ]
177
190
210
232
240
253
273
280
312
330
345
357
385
408
462
520
760
840
1320
62
Lattice sums
Table 1.6 (cont.)
c
(i) a = 1, b = 0
1365
2−3 [L +1 L −5460 + L −3 L +1820 + L +5 L −1092 + L −7 L +780 + L +13 L −420 + L −15 L +364
+ L +21 L −260 + L −35 L +156 + L −39 L +140 + L +65 L −84 + L −91 L +60 + L +105 L −52
+ L −195 L +28 + L +273 L −20 + L −455 L +12 + L −4 L +1365 ]
2−3 [(1 − 2−s + 21−2s )L +1 L −1848 + L −4 L +1848 + (1 + 2−s + 21−2s )L −3 L +616
+ L +12 L −616 + (1 − 2−s + 21−2s )L −7 L +264 + L +28 L −264 + (1 + 2−s + 21−2s )
× L −11 L +168 + L +44 L −168 + (1 + 2−s + 21−2s )L +21 L −88 + L −84 L +88
+ (1 − 2−s + 21−2s )L +33 L −56 + L −132 L +56 + (1 + 2−s + 21−2s )L +77 L −24
+ L −308 L +24 + (1 − 2−s + 21−2s )L −231 L +8 + L −8 L +924 ]
1848
c
(ii) a = 1, b = 1
1
2
3
4
5
7
9
11
13
17
19
23
25
29
31
37
41
47
49
59
67
79
6L +1 L −3
2L +1 L −7
2L +1 L −11
L +1 L −15 + L −3 L +5
2L +1 L −19
2(1 − 2 × 3−s + 31−2s )L +1 L −3
L +1 L −35 + L +5 L −7
2L +1 L −43
L +1 L −51 + L −3 L +17
2L +1 L −67
(1 + 51−2s )L +1 L −3 + L +5 L −15
L +1 L −91 + L −7 L +13
(1 − 2 × 3−s + 31−2s )L +1 L −11 + L −3 L +33
L +1 L −115 + L +5 L −23
L +1 L −123 + L −3 L +41
(1 − 2 × 7−s + 71−2s )L +1 L −3 + L −7 L +21
2L +1 L −163
L +1 L −187 + L −11 L +17
2−1 [L +1 L −195 + L −3 L +65 + L +5 L −33 + L −15 L +13 ]
L +1 L −235 + L +5 L −47
L +1 L −267 + L −3 L +89
2−1 [(1 − 2 × 3−s + 31−2s )L +1 L −35 + L −3 L +105 + (1 + 2 × 3−s + 31−2s )L +5 L −7
+ L −15 L +21 ]
L +1 L −403 + L +13 L −31
L +1 L −427 + L −7 L +61
2−1 [L +1 L −435 + L −3 L +145 + L +5 L −87 + L −15 L +29 ]
2−1 [L +1 L −483 + L −3 L +161 + L −7 L +69 + L −23 L +31 ]
2−1 [L +1 L −555 + L −3 L +185 + L +5 L −111 + L −15 L +37 ]
2−1 [L +1 L −595 + L +5 L −119 + L −7 L +85 + L +17 L −35 ]
2−1 [L +1 L −627 + L −3 L +209 + L −11 L +57 + L −19 L +33 ]
2−1 [L +1 L −715 + L +5 L −143 + L −11 L +65 + L +13 L −55 ]
2−1 [L +1 L −795 + L −3 L +265 + L +5 L −159 + L −15 L +53 ]
2−2 [L +1 L −1155 + L −3 L +385 + L +5 L −231 + L −7 L +165
+ L −15 L +77 + L +21 L −55 + L +33 L −35 + L −11 L +105 ]
2−1 [L +1 L −1435 + L +5 L −287 + L −7 L +205 + L −35 L +41 ]
2−2 [L +1 L −1995 + L −3 L +665 + L +5 L −399 + L −7 L +285
+ L −19 L +105 + L −15 L +133 + L +21 L −95 + L −35 L +57 ]
2−2 [L +1 L −3003 + L −3 L +1001 + L −7 L +429 + L −11 L +273
+ L +13 L −231 + L +21 L −143 + L +33 L −91 + L −39 L +77 ]
2−2 [L +1 L −3315 + L −3 L +1015 + L +5 L −663 + L +13 L −255
+ L +17 L −195 + L −15 L +221 + L −39 L +85 + L −51 L +85 ]
101
107
109
121
139
149
157
179
199
289
359
499
751
829
1.5 Contour integral technique
63
1.5 Contour integral technique
We have seen that many lattice sums can be expressed as hyperbolic double sums.
It appears that for several of these there exist representations as contour integrals.
In at least one case the integral can be evaluated by residues and the lattice sum
obtained in closed form, as we shall now illustrate.
Consider the integral
#
π z 2l dz
1
(l = 0, 1, 2, . . .),
Sl (n) =
2πi C (z 2 + n 2 )1/2 sinh π(z 2 + n 2 )1/2 sin π z
(1.5.1)
where C is the simple closed curve shown in Fig. 1.4. Note that the integrand has
simple poles at z = ± p, ±i( p 2 +n 2 )1/2 , the where p = 0, 1, 2, . . .. By evaluating
the integral with respect to the residues at poles lying within C, we obtain
Sl (n) =
1 (−1)l n 2l−1
δl,0
−
.
2n sinh (π n) 2 sinh (π n)
(1.5.2)
However, if we view C as a closed curve surrounding the remaining poles, we find
Sl (n) = (−1)l
∞
(−1) p ( p 2 + n 2 )(l−1/2) cosech π( p 2 + n 2 )1/2
p=1
−
∞
p=1
(−1) p p 2l
cosech π( p 2 + n 2 )1/2 .
( p 2 + ln 2 )1/2
(1.5.3)
On combining (1.5.2) and (1.5.3) we have
∞
p=1
(−1) p
[ p 2l + (−1)l+1 ( p 2 + n 2 )l ]
1 (−1)l n 2l−1
(1 − δl,0 ). (1.5.4)
=
2 sinh (π n)
( p 2 + n 2 )1/2 sinh π( p 2 + n 2 )1/2
i √ n2 + 1
i|n|
C
−2
−1
1
2
−i|n|
−i √n2 + 1
Figure 1.4 Contour for the integration in Equation (1.5.1).
64
Lattice sums
Now, if F(n) is any function whatsoever, we have the identity
∞ ∞
(−1) p F(n)
[ p 2l + (−1)l+1 ( p 2 + n 2 )l ]
( p 2 + n 2 )1/2 sinh π( p 2 + n 2 )1/2
=
n 2l−1 F(n)
1
(−1)l (1 − δl,0 )
.
2
sinh (π n)
n=1 p=1
∞
(1.5.5)
n=1
It is clear that a number of identities of this sort may be derived in a similar
fashion, but we have not yet explored this to any great extent.
When the function F(n) is a polynomial, the sums which occur on the righthand side of (1.5.5) were studied by Zucker [146]. Thus, if we take l = 1, F(n) =
(−1)n+1 , we obtain
∞ ∞
(−1) p+n
n=1 p=1
(2 p 2 + n 2 )
1
.
=
8π
( p 2 + n 2 )1/2 sinh π( p 2 + n 2 )1/2
(1.5.6)
By interchanging p and n and adding the resulting sum to (1.5.6) there results
∞ ∞
(−1) p+n
n=1 p=1
( p 2 + n 2 )1/2
1
.
=
2
2
1/2
12π
sinh π( p + n )
(1.5.7)
In the calculation of the electric field gradient at an ion site in a body-centred
cubic lattice one requires the sum
P2 (cos θ )
σ =
,
(1.5.8)
R3
where θ is the angle between the lattice vector R and the z-axis and the sum is over
the sublattice composing the structure that does not contain the site in question.
This series is conditionally convergent, but by planewise summation de Wette [37]
has shown that it can be put into the form
σ = 2π 2
(−1) p+n
( p 2 + n 2 )1/2
,
sinh π( p 2 + n 2 )1/2
(1.5.9)
where the sum is over all values of ( p, n) = (0, 0). This is easily expressed in
terms of (1.5.7) and the sum
∞
(−1)n n
n=1
sinh π n
=−
1
,
4π
(1.5.10)
whence we have
σ =−
4π
= −4.188790205 . . . ,
3
(1.5.11)
whereas de Wette’s numerical result is σ = −4.189. It appears to us that this procedure is capable of being greatly extended and may lead to the exact evaluation
1.6 Conclusion
65
of many other useful lattice sums. A completely different class of contour integral
representation was obtained by Glasser [56] using the Bromwich integral form
for the inverse Mellin transform applied to the theta function representations discussed in Section 1.3. It will be interesting to see whether these integrals can be
applied directly to the problem.
1.6 Conclusion
The topic of lattice sums has roots and branches in widely differing areas of chemistry, physics, and mathematics. From a historical perspective, the subject appears
as one having few themes but many variations. Of these we have attempted only
to ring the changes on those themes which seem to us to be the most promising for future development and application. Thus we have discussed in some
detail: (a) the θ -function approach, which subsumes the usual Ewald method;
(b) the number-theoretic approach, which leads to many interesting and beautiful
results whose utility is yet to be fully exploited. We have also briefly outlined
the use of contour integration, which appears to offer a convenient and flexible representation of lattice sums and which has not yet been systematically
explored.
The following questions are deserving of further investigation.
(1) Is it possible to develop alternative methods of systematically generating qseries which lie outside the realm of θ -functions? One of us (Glasser [59])
has found exactly a five-dimensional sum using q-series stemming from the
Heine generalization of the hypergeometric function.
(2) Is the conjecture (Glasser [57]) with regard to two-dimensional lattice sums
correct? For, although we can only solve exactly for s = 1 those lattice sums
to which the conjecture applies for all s, it is possible to find S1 (a, b, c),
S2 (a, b, c), and S1,2 (a, b, c) exactly in terms of π and logarithms of surds
for any a, b, and c (Zucker [145]).
(3) Will the contour integral approach lead to simpler methods for calculating
lattice sums?
(4) Is there some connection between lattice sums and the well-known Watson
integrals [138] for the corresponding lattice?
We have not even touched on many-body lattice sums, which are of growing
interest in chemistry and physics, nor on spatially restricted sums (Hoskins et al.
[78]; Stepanets et al. [128]).
We conclude with Samuel Johnson that ‘Nothing amuses more harmlessly
than computation and nothing is more applicable to real business and speculative
inquiry’.
66
Lattice sums
1.7 Appendix: Complete elliptic integrals in terms
of gamma functions
We wish here to outline the application of our results on two-dimensional sums
to an old problem concerning complete elliptic integrals. Namely: when can K be
expressed in terms of gamma functions for rational arguments? For this purpose,
we consider the sums
2
(m + a 2 n 2 )−s ,
S0 (a, s) =
(−1)m (m 2 + a 2 n 2 )−s ,
S1 (a, s) =
(1.7.1)
(−1)n (m 2 + a 2 n 2 )−s ,
S2 (a, s) =
(−1)m+n (m 2 + a 2 n 2 )−s ,
S3 (a, s) =
over all pairs of integers (m, n) = (0, 0). It has been shown by two of the present
authors (Glasser [58], Zucker [145]) that
2θ 3 (π α, q) 2
exp [2πi(nx + my)]
2π
1
2 y
(1.7.2)
ln = 2π
−
,
b
3|ab| θ1 (0, q) a 2 n 2 + b2 m 2
where q = exp (−π |a/b|), α = x + i y|a/b|, and thus
4k 2
π
S1 (a, 1) = −
ln
,
3|a|
k
2
4k
π
,
ln
S2 (a, 1) = −
3|a|
k
4
π
,
ln
S3 (a, 1) = −
3|a|
kk (1.7.3)
where k 2 + k 2 = 1, K (k ) = a K (k). Hence, when one of the Si (a, s) is solvable, k is expressible in terms of radicals. Furthermore, we have Kronecker’s
Grenzformel (Selberg and Chowla [124])
4|a| k k 1/3
2π γ
2π
π
=
−
ln
K . (1.7.4)
lim S0 (a, s) −
s→1
a(s − 1)
|a|
|a|
π
4
Hence, when S0 and another Si are solvable, K can be expressed in closed form,
as is illustrated by the following example.
In the case a = 1, we have
S0
S3
= 4β(s)ζ (s),
= −4(1 − 21−s )ζ (s)β(s).
(1.7.5)
From
√ this we obtain S3 (1, 1) = −π ln 2 = −(π/3) ln (4/kk ). Therefore k =
1/ 2. From (1.7.4), we find
lim
s→1
S0 −
π
s−1
= 2π γ +
2( 34 )
π
ln 2 − π ln 2 + π ln
2
( 14 )
4
.
(1.7.6)
1.8 Appendix: Watson integrals
67
√
√
From (1.7.6) we easily see that K (1/ 2) = 2 ( 14 )/4 π . For further details and
a table of similar results, see Zucker [145].
1.8 Appendix: Watson integrals
The generating functions for random walks on a lattice, along with a number of
other ‘propagators’ for lattice phenomena, can be cast in the form
W =
dk
.
z − φ(k)
(1.8.1)
The integration in (1.8.1) is over the Brillouin zone and φ is a trigonometric
polynomial in the components of k. Because G. N. Watson [138] was the first
to evaluate an important family of these, they are now called Watson integrals.
A great deal of effort has gone into evaluating these integrals for various lattices,
much of which was summarized by Glasser and Zucker [64]; see also Hioe [76].
One particularly striking result is (Joyce [86], [87]) the following:
I =
=
k± =
ξ=
1
π3
π π
0
0
π
0
du dv dw
z − cos u − cos v − cos w
(ξ + 1)(ξ + 4)
8
2
2
π z ξ + 8ξ + 8 + 4(2 + ξ )(ξ + 1)1/2
1/2
K (k+ ) K (k− ),
(1.8.2)
ξ [(ξ + 1)1/2 ± (ξ + 4)1/2 ] − 2[1 − (ξ + 1)1/2 ]
,
ξ [(ξ + 1)1/2 ± (ξ + 4)1/2 ] + 2[1 − (ξ + 1)1/2 ]
z 2 + 3 − [(z 2 − 9)(z 2 − 1)]
;
z 2 − 3 + [(z 2 − 9)(z 2 − 1)]1/2
this result is for the simple cubic lattice. For the case z = 3 we find Watson’s
formula
√
√
√
√ √
√ √
8
[10(17 − 12 2)]1/2 K [(2 − 3)( 3 + 2)] K [(2 − 3)( 3 − 2)] .
2
3π
(1.8.3)
By means of the sums for a 2 = 6 the method in Appendix 1.7 can be applied to
evaluate the elliptic integrals in (1.8.3), and we obtain
I =
√
I =
6
5
7
( 1 )( 24
)( 24
)( 11
24 ).
96π 3 24
This raises the interesting question: is there a direct connection between lattice
sums and Watson integrals for a given lattice?
68
Lattice sums
1.9 Commentary: Watson integrals
In this commentary we sketch the method of determination of the classical Watson
integrals and make some further elaborations on the previous appendix. More
details on Watson integrals can be found in Chapter 9.
1.9.1 The easiest three-dimensional Watson integral
We start with the easiest integral to evaluate. Let
π π π
1
d x d y dz,
W3 (w) =
0
0
0 1 − w cos x cos y cos z
for suitable w > 0. One may prove that
π π π
1
d x d y dz
W3 (1) =
1 − cos x cos y cos z
0
0
0
= 14 4 14 = 4π K √1 ,
2
via the binomial expansion and [24, Exercise 14, p. 188].
More generally (see below),
W3 ((2kk )2 ) = π 3 3 F2
1 1 1
2, 2, 2
1, 1
; 4 k 2 1 − k 2 = 4π K 2 (k) .
1.9.2 The harder three-dimensional Watson integrals
We now describe results found largely in Joyce and Zucker [91, 92], where more
background can also be found. The following integral arises in Gaussian and
spherical models of ferromagnetism and in the theory of random walks.
One of the most impressive closed-form evaluations of a multiple integral is that
of Watson:
π π π
1
d x d y dz
W1 =
−π −π −π 3 − cos x − cos y − cos z
1 √
1
=
( 3 − 1) 2 24
2 11
(1.9.1)
24
96 √
√
√ 2
= 4π 18 + 12 2 − 10 3 − 7 6 K (k6 ) ,
√ √ √
2− 3
3 − 2 is the sixth singular value; that is,
√
K (k6 )/K (k6 ) = 6. Elliptic integrals at the singular values may be evaluated
in term of gamma functions;
$ ∞ a large number of cases were already known in [54].
computation [91]
Note that W1 = π 3 0 exp(−3t) I03 (t) dt allows for efficient
$π
here the Bessel function I0 (t) has been written as (1/π ) 0 exp[t cos θ ] dθ . The
evaluation (1.9.1), in its original form, is due to Watson and is a tour de force.
where k6
=
1.9 Commentary: Watson integrals
69
Next we describe a refined and simplified evaluation due to Joyce and Zucker
[92]. Similarly, the integral
π π π
d x d y dz
W2 =
0
0
0 3 − cos x cos y − cos y cos z − cos z cos x
√
% &
2
1
= 3π K (sin( 12
(1.9.2)
π )) = π4 2−2/3 β 2 13 , 13 ,
1
where sin 12
π = k3 is the third singular value. Indeed, as we shall see, (1.9.2)
is easier and can be derived on the way to (1.9.1).
The evaluation (1.9.2) then implies that
π π
1
dy dz
W2 =
2
2
2
2 1/2
π
0 0 (9 − 8 cos y cos z − cos y − cos z + cos y cos z)
after careful performance of the innermost integration. The expression inside the
square root factors as (cos x cos y + cos x + cos y − 3)(cos x cos y − cos x −
cos y − 3). Upon substituting s = x/2, and t = y/2, one obtains
π/2 π/2
dy dx
%
&%
&
2
2
[ 1 − sin x sin y 1 − cos2 x cos2 y ]1/2
0
0
∞ ∞
m + n √
1 3
1
1
1
= 3 K 2 sin( 12
=
β n + 2, m + 2
π) .
4π
n
m=0 n=0
1.9.3 More about the Watson integrals
For a > 1, b > 1, one can show that
1
2
=
'
2(b+a)
K
(1+b)(1+a)
dy
= √
1/2
(1 + b) (1 + a)
0 [(a + cos y) (b − cos y)]
1/2 π
dt
π
[(1 + b) (1 + a) cos2 t + (1 − a) (1 − b) sin2 t]1/2
0
.
A beautiful, but harder to establish, identity is
(
(
π/2 1−c
1+c
2
2
2
K
c cos s + sin s ds = K
K
,
2
2
0
(1.9.3)
or equivalently
π/2
K
with
k
=
% &
1 − (2kk )2 cos2 θ dθ = K (k) K k 0
√
1 − k 2 . Hence,
π/2 '
√
K
1 − (2k N k N )2 cos2 θ dθ = N K 2 (k N ) ,
0
70
Lattice sums
where k N is the N th singular value. This is especially pretty for N = 1, 3, 7;
2 k N k N = 1, 12 , 18 , respectively.
One may deduce that the face-centred cubic (FCC) lattice for the Green’s
function evaluates as
π/2 '
√
1
3
2 s + sin2 s ds = 3 K 2 (k ) .
W2 =
K
cos
3
4
π
0
Correspondingly, Watson’s evaluation for the simple cubic (SC) lattice relied on
deriving
π √
cos x − 5
d x,
W1 = 2 π
K
2
0
and the following extension of (1.9.3):
π/2 K
c2 cos2 s + d 2 sin2 s ds
0
)
)
1 − cd − (d 2 − 1)(c2 − 1)
1 + cd − (d 2 − 1)(c2 − 1)
K
.
=K
2
2
1.9.4 The generalized Watson integrals
Let
W1 (w1 ) =
W2 (w2 ) =
π π π
d x d y dz
−π −π −π 3 − w1 (cos x − cos y − cos z)
π π π
d x d y dz
0
0
,
0 3 − w2 (cos x cos y − cos y cos z − cos z cos x)
.
In a beautiful study, Joyce and Zucker [92], using the type of elliptic and
hypergeometric transformations that we have explored, showed fairly directly that
W2 (−w1 (3 + zw1 )/(1 − w1 )) = (1 − w1 )1/2 W1 (w1 ).
It is now easy to verify that, with w1 = −1, this leads to a quite direct evaluation
of (1.9.1) from (1.9.2).
It is also true that
π/2 √
K 12 + 12 sin2 t dt.
W1 = 2 π
0
One may also give a more symmetric form, namely
π/2 '
%
&
K
1 − 4k 2 1 − k 2 cos2 x d x
=
0
for 0 < k < 1.
0
π/2 π/2
0
dt d x
[cos2 t
+ 4k 2 (1 − k 2 ) cos2
x sin2 t]1/2
1.9 Commentary: Watson integrals
71
Hint: For (1.9.3) consider N = 3 (c2 = 34 ) in (1.9.3), and let a and b be defined
as a = (3 − cos x) / (1 + cos x) and b = (3 + cos x) / (1 − cos x) in (1.9.3).
1.9.5 Watson integral and Burg entropy
Consider the perturbed Burg entropy maximization problem
1 1 1
v (α) := sup log ( p(x1 , x2 , x3 )) p(x1 , x2 , x3 ) d x1 d x2 d x3 = 1
0
0
0
p≥0
1 1 1
and, for k = 1, 2, 3,
p(x1 , x2 , x3 )
0
0
0
× cos (2π xk ) d x1 d x2 d x3 = α ,
maximizing the log of a density p with given mean and with the first three
cosine moments fixed at a parameter value 0 ≤ α < 1. It transpires that there
is a parameter value α such that below and at that value v(α) is attained while
above that value it is finite
$ but unattained. This is interesting, because the general
method – maximizing T log ( p(t)) dt subject to a finite number of trigonometric
moments – is frequently used. In one or two dimensions, such spectral problems
are always attained when feasible.
There is no easy way to see that this problem qualitatively changes at α, but we
can get an idea by considering
p (x1 , x2 , x3 ) =
1/W1
3
3 − 1 cos (2π xi )
and checking that this is feasible for
α = 1 − 1/(3W1 ) ≈ 0.340537329550999142833;
here α is expressed in terms of the first Watson integral, W1 .
By using Fenchel duality [28] one can show that the above p is optimal. Indeed,
for all α ≥ 0 the only possible optimal solution is of the form
p α (x1 , x2 , x3 ) =
λ0α
−
3
1
i
1 λα
cos (2π xi )
,
for some real numbers λiα . Note that we have four coefficients to determine; using
the four constraints we can solve for them. For 0 ≤ α ≤ α the precise form is
parameterized by the generalized Watson integral:
p α (x1 , x2 , x3 ) =
1/W1 (w)
,
3
3 − 1 w cos (2π xi )
and α = 1 − 1/[3W1 (w)],
from zero to one. Note also that the
$ ∞ as3 w ranges
3
−3t
expression W1 (w) = π 0 I0 (w t) e dt allows one to quickly obtain w from
72
Lattice sums
α numerically. For α > α, no feasible reciprocal polynomial can stay positive.
Full details are given in [25].
1.10 Commentary: Nearest neighbour distance
and the lattice constant
The ‘size’ of the Madelung constant for any given crystal depends on whether
one refers to the nearest neighbour distance or to the lattice constant. In the salt
lattice, the nearest neighbour distance, that is, the distance between a sodium ion
and a chlorine ion, is exactly half the distance between neighbouring sodium ions,
which is the lattice constant. One will find both values scattered around in the
literature. So, for NaCl, in terms of the nearest neighbour distance the Madelung
constant is 1.74756. . . and in terms of the lattice constant it is 3.49513 . . . In
Table 1.2 listing Madelung constants, all are given in terms of the lattice constant.
In the early days of Madelung constants, when one had only very simple crystals with two or at most three ions to deal with, the nearest neighbour distance
was the preferred choice, but when more complex crystals such as perovskite and
spinel structures – which have large numbers of ions – were investigated, the
lattice constant seemed a more sensible distance to refer to.
1.11 Commentary: Spanning tree Green’s functions
On any graph G a spanning tree is a connected loop-free subgraph containing all
the points (vertices) of G. It was proved by Temperley and others that if G is a
regular lattice L restricted to N sites then the number of spanning trees TN (G)
grows exponentially as N increases without bound and that the limit
1
log TN (G),
N
called the spanning tree constant for L, exists and is positive. Furthermore,
1
λL = d
log[w − (θ )] d d θ,
(1.11.1)
π [0,π ]d
λ L = lim
N →∞
where is the structure function for L and w is the smallest positive (rational)
value for which the integrand in (1.11.1) is singular.
The spanning tree constant was first investigated for one- and two-dimensional
lattices by Temperley [130]. Lin [99], Wu [140], Glasser and Wu [63], Chen and
Wu [36], and Glasser and Lamb [62] evaluated the integral
2π 2π
1
log A + B + C − A cos θ1 − B cos θ2 − C cos(θ1 + θ2 ) dθ1 dθ2 ,
4π 2 0
0
(1.11.2)
which applies to the square and triangular lattices among others.
1.11 Commentary: Spanning tree Green’s functions
73
Higher-dimensional spanning tree constants were discussed by Shrock and Wu
[127], who found, for a d-dimensional body-centred cubic lattice,
∞
1
1 1 (2l)! d
(d)
d
log[1 − cos θ1 · · · cos θd ] d θ =
,
λ BCC = − d
π [0,π ]d
2
l 22l l!2
l=1
and by Joyce [88], who independently showed, for d = 3, that
π π π
1
(3)
log(1 − cos θ1 cos θ2 cos θ3 ) dθ1 dθ2 dθ3
λ BCC = − 3
π 0 0 0
=
1
5 F4
16
3 3 3
2 , 2 , 2 , 1, 1 ; 1
2, 2, 2, 2
.
(1.11.3)
Here, we have used the generalized hypergeometric function, defined by
∞
(a1 )n (a2 )n · · · (a p )n z n
a1 , a2 , . . . , a p
;z =
,
p Fq
b1 , b2 , . . . , bq
(b1 )n (b2 )n · · · (bq )n n!
n=0
where (a) p = (a + p)/ (a) is the Pochhammer symbol.
Though it arises in a different context, we also mention the Baxter–Bazhanov
formula (1993, unpublished)
1
log(2 − cos θ1 − cos θ2 − cos θ3 + cos θ1 cos θ2 cos θ3 ) d 3 θ
λB B = 3
π [0,π ]3
8G
=
− 3 log 2,
(1.11.4)
π
where G denotes Catalan’s constant.
If we write L = J(w) and treat w as a complex parameter then
dd θ
1
J (w) = d
(1.11.5)
π [0,π ]d w − (θ)
is the generalized Watson integral (Green’s function) for L, and thus the function
J (w) is of considerable interest. Other than for d = 2, only a few closed-form
evaluations are available, such as the Baxter–Bazhanov case studied by Joyce et
al. [90]. However, this matter is closely tied in with the recent surge of interest in
the logarithmic Mahler measure, defined by
log P(e2πit1 , . . . , e2πitn ) d n t,
(1.11.6)
m(P) =
[0,1]n
where P is an n-variate Laurent polynomial. For example, the value of
m(4 − x − 1/x − y − 1/y),
equivalent to a series identity due to Ramanujan, gives
π π
1
4G
log(2 − cos θ1 − cos θ2 ) dθ1 dθ2 =
λsq = 2
− log 2.
π
π 0 0
(1.11.7)
74
Lattice sums
Numerous conjectured and established values are available for a variety of polynomials, in the work of Boyd [30], Rogers [117], Bertin [13], and others. Several
important three-dimensional cases have been announced recently. Thus a study of
the Mahler measure by Joyce [89] relevant to the FCC lattice led to a formula for
J (w) for the FCC case, and in the limit w → 3 he obtained
λ FCC = 2λdiam
π π π
1
= 3
log(3 − cos θ1 cos θ2 − cos θ2 cos θ3 − cos θ3 cos θ1 ) dθ1 dθ2 dθ3
π 0 0 0
=−
14
8
8
log 2 + log 3 −
5 F4
15
5
135
4 3 5
3 , 2 , 3 , 1, 1 ; 1
2, 2, 2, 2
.
(1.11.8)
Guttmann and Rogers [66, 67] and Glasser [60] found J (w) for both the Baxter–
Bahzanov case and the simple cubic case. For the simple cubic they obtain (by
using modular parameterizations for hypergeometric functions)
e−x − e−wx I03 (x)
dx
x
0
R13
1
9 R13
1, 1, 54 , 32 , 74
2R0
=
+
;
16
F
log
4
5
5
8 R24
2, 2, 2, 2
(w 2 + 3)3
R24
1, 1, 54 , 32 , 74
R1
3 R1
; 16 4
+
,
(1.11.9)
5 F4
4
8 R3
2, 2, 2, 2
R3
JSC (w) =
∞
where
√
R0 = w 3 − 5w − (w 2 − 1) w 2 − 9,
√
R3 = w 2 − 9 − w w 2 − 9,
√
R1 = 2w 2 − 9 − 2w w 2 − 9,
√
R4 = w 2 − 3 − w w 2 − 9.
A more direct connection between various Mahler measures and lattice sums is
touched on in Chapter 6.
1.12 Commentary: Gamma function values in terms
of elliptic integrals
Since the first appearance of the work presented in this chapter the material of
Appendix 1.7 has been considerably extended and the inverse question has been
addressed, in [29]. This inverse question has complexity-theoretic implications,
as it allows certain gamma and beta function values – such as π – to be computed as rapidly as complete elliptic integrals [23]. More recently, in 2008, Zucker
and Broadhurst in previously unpublished computations provided the remarkable
results in singular value theory given below.
1.12 Commentary: Gamma function values in terms of elliptic integrals 75
Let K (k) be the complete elliptic integral of the first kind, given by
1
dx
K (k) :=
,
2
0
(1 − x )(1 − k 2 x 2 )
(1.12.1)
where k denotes the modulus. For a given positive integer N , consider the equation
K (k) √
= N,
(1.12.2)
K (k)
where K (k) ≡ K (k ) and k ≡ (1 − k 2 )1/2 is the complementary modulus.
When k ∈ (0, 1), it is known that (1.12.2) has a unique solution k ≡ k N which
is an algebraic number – often, but not always soluble in radicals; these kn are
the so-called singular values [24]. As also described in [24], Selberg and Chowla
[123, 124] proved that K (k N ) ≡ K [N ] could be expressed in terms of products
of functions, algebraic numbers, and powers of π . Borwein and Zucker [29]
showed that these formulae were much simplified if they were expressed in terms
of the Euler beta function B( p, q), given by
B( p, q) =
( p)(q)
.
( p + q)
(1.12.3)
Further, for all N = 1, 2 (mod 4) an extra simplification was achieved, since it
was found that only the central beta function, given by
β( p) := B( p, p) =
2 ( p)
,
(2 p)
(1.12.4)
was needed to express K [N ].
In every case the K [N ] become products of beta functions and algebraic numbers only. In the cases N = 1, 2 (mod 4) it was also found that β’s always appear
in pairs β( p)β( 12 − p). So, making use of the relation
β( 12 − p) = 24 p−1 tan(π p)β( p),
(1.12.5)
one could always reduce the argument of the β( p)’s involved so that p ≤ 14 .
One really only needs to give K [N ] for N square-free. For, suppose that N =
n 2 μ where n and μ are positive integers; then it is known that
K [n 2 μ] = Mn (μ)K [μ],
(1.12.6)
where Mn (μ) is algebraic and is called the multiplier [24]. The following
formulae for small Mn (μ) are known:
M2 (μ) =
1 + kμ
2
27M34 (μ) − 18M32 (μ) − 8(1 − 2kμ2 )M3 (μ) − 1 = 0,
(5M5 (μ) − 1) (1 − M5 (μ)) =
5
,
(1.12.7)
(1.12.8)
256kμ2 (1 − kμ2 )M5 (μ).
(1.12.9)
Many more multipliers are given in [24].
76
Lattice sums
These formulae for finding K [4N ], K [9N ], and K [25N ] from K [N ] depend
only on knowing the singular value k N [24]. Thus, suppose we wish to find K [25]
from K [1]. We substitute the value k12 = 1/2 into (1.12.9).
In this particular
√
case
√ (1.12.9) can be solved, giving the result M5 = ( 5 + 2)/5, so K [25] =
( 5 + 2)K [1]/5.
In compiling this compendium in appendix section A.3 at the back of the book,
Zucker in 2005 first found (where possible) the values of kμ2 for 1 ≤ μ ≤ 25.
Many of these results were obtained using the Mathematica routine ‘Recognize’.
It will be seen that, for large N , associated with each β(n) is a factor tan(π n).
Actually this is so for every N which is expressed in terms of central β functions,
but for small N the tangents can be found in terms of radicals. In fact for 1 ≤
N ≤ 20, excluding N = 11 and 19, all K [N ] can be expressed in quadratic
surds and at most three central beta functions. The most surprising of these is the
case N = 17. Subsequently, in 2008, Broadhurst and Zucker extended the data
in appendix section A.3 to include all K [N ] for 1 ≤ N ≤ 100, and this exercise
required us to refine some views expressed above.
(1) It became obvious that the best way to express the results most concisely
√
was to use the function b( p) = β( p) tan( pπ ).
(2) Because of a remarkable relation found empirically by David Broadhurst,
it was possible to express every K [N ] in terms of the b( p) and algebraic
quantities. It was well known that, for square-free N , the transcendental
part of K [N ] contains the quantity
G(D) :=
|D|−1
k=1
D
k (k )
,
|D|
where
% D & D = −4N if N ≡ 1, 2 (mod 4), D = −N if N ≡ 3 (mod 4), and
k is the Legendre–Jacobi–Kronecker symbol.
(3) Broadhurst discovered the beautiful general formula
2−( D)
2
(2π )h(D) G(D)
= B(D)2 ,
B(D) =
(|D|−1/k)
k=1
b
D
k (k )
,
|D|
where h(D) is the class number of the binary quadratic form am 2 + bmn +
cn 2 with fundamental discriminant D = (b%2 −
& 4ac) < 0. For square-free
N ≡ 1, 2 (mod 4), D = −4N is even and D2 = 0. For square-free N ≡
3 (mod
N ≡ 3 (mod 8) with
&
% D & 4), D = −N and two cases arise,% Dnamely
2 − 2 = 3 and N ≡ 7 (mod 8) with 2 − 2 = 1.
Zucker’s method for finding K [N ] using disjoint and semi-disjoint discriminants
was not available for most of the N values missing from the original list, but
Broadhurst, using much more sophisticated numerical techniques, extracted all
1.13 Commentary: Integrals of elliptic integrals, and lattice sums
77
the required results. It seems that for N ≡ 2 (mod 4), K [N ]8 k 3N (1 − k 2N ) plays a
pivotal role, with the invariants
P[N ] :=
2k N
,
1 − k 2N
used by Klein [24], or
Q[N ] := P[N ]1/3 +
1
P[N ]1/3
being most easily found in radicals. For N ≡ 1 (mod 4), K [N ]8 k 2N (1 − k 2N )
appears as a feature, with either the quantity
V [N ] := 4k 2N (1 − k 2N )
or the quantity
W [N ] =: V [N ]1/3 +
1
V [N ]1/3
being more easily expressible in radicals. Also, for convenience, let
T [N ] := k 3N (1 − k 2N ).
Broadhurst has conjectured that the algebraic part of K [N ] can always be found
in radicals despite the algebraic equations involved being of high degree. Indeed,
equations of degree as high as 23 and 25 have been radicalized.
For a list of K [N ], see appendix section A.3.
1.13 Commentary: Integrals of elliptic integrals,
and lattice sums
It is of some interest to consider integrals of the complete elliptic integrals E and
K . Since the derivatives of E and K are again elliptic integrals, one may use
integration by parts, for instance, to find some indefinite integrals, such as
x K (x) d x = E(x) + (x 2 − 1)K (x).
A list of definite integrals involving K was compiled by Glasser [61], while a
systematic study was that of Wan [136], where integrals of products such as K (x)2
are given in terms of hypergeometric functions.
Evaluations of this kind, as well as neat identities like
1
1
K (x)2 d x =
K (x)2 d x,
2
0
0
78
Lattice sums
occur naturally in the study of uniform random walks in the plane (i.e., not
on a lattice) and in Mahler measures; see [26, 27]. In [136], it was observed
experimentally that
1
8 14
K (x)3 d x =
.
(1.13.1)
128π 2
0
Closed-form evaluations of the integral of the cube
function, such as
$ 1 of a special
3 d x), are expected to be
K
(x)
(1.13.1) and its many equivalent forms (e.g., 10
3 0
rare. A recent work by Rogers, Wan, and Zucker [118] gives a proof of (1.13.1).
We start with the parametrizations (see [24]), with log q = −π K (k)/K (k),
K (k) =
π 2
θ (q),
2 3
k=
θ22 (q)
θ32 (q)
k =
,
θ42 (q)
θ32 (q)
,
k 4
dk
=
θ (q).
dq
2q 4
Then we can write the integral in (1.13.1) as
1
I =−
0
log3 q 2
θ (q)θ34 (q)θ44 (q) dq.
16q 2
Using a result of Ramanujan and η-function transformations, the integral can be
written as
1
log3 q 4
η (q)η2 (q 2 )η4 (q 4 ) dq.
I = −5
q
0
Now, with q = e2πiτ , f (τ ) = η4 (τ )η2 (2τ )η4 (4τ ) is a weight-5 modular form,
which has the Fourier series
f (τ ) =
∞
an e2πinτ ,
n=1
and therefore the last integral is the following L-series (as can be seen by taking
a Mellin transform):
I = 30
∞
an
n=1
n4
.
(This is known as a central, or critical, L-value of f .) It can be shown by standard
methods (see e.g., Section 5.1) that in fact
f (τ ) =
∞
1 2
2
(n − im)4 q n +m ,
4 m,n=−∞
References
79
and so our integral reduces to
I =
15 (n − im)4
1
15 =
,
2
2
4
2 m,n (n + m )
2 m,n (n + im)4
√
which, from a computation in Section 4.8, is 2K (1/ 2)4 . Equation (1.13.1) now
follows.
The method outlined here can be used to establish connections between classes
of integrals of E and K and lattice sums. Sometimes the integrals can be evaluated
independently, therefore providing new results for lattice sums. Three examples
are given here:
(−1)m+1
m,n
(−1)m+n+1
m,n
m,n
(−1)m+1
2 ( 18 ) 2 ( 38 )
m 2 − 2n 2
,
=
48π
(m 2 + 2n 2 )2
6 ( 13 )
m 2 − 3n 2
=
,
14
(m 2 + 3n 2 )2
2 3 π2
4 ( 14 )
m 2 − 4n 2
=
.
32π
(m 2 + 4n 2 )2
For more integrals involving K , see Chapter 9. We note that (1.13.1), among
other integrals, has also been proved recently and independently in Zhou [141].
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2
Convergence of lattice sums
and Madelung’s constant
The lattice sums involved in the definition of Madelung’s constant for an
NaCl–type crystal lattice in two or three dimensions are investigated here. The
fundamental mathematical questions of convergence and uniqueness of the sum
of these series, which are not absolutely convergent, are considered. It is shown
that some of the simplest direct sum methods converge and some do not converge. In particular, the very common method of expressing Madelung’s constant
by a series obtained from expanding spheres does not converge. The concept of
the analytic continuation of a complex function to provide a basis for an unambiguous mathematical definition of Madelung’s constant is introduced. By these
means, the simple intuitive direct sum methods and the powerful integral transformation methods, which are based on theta function identities and the Mellin
transform, are brought together. A brief analysis of a hexagonal lattice is also
given.
2.1 Introduction
Lattice sums have played a role in physics for many years and have received a
great deal of attention on both practical and abstract levels. The term ‘lattice sum’
is not a precisely defined concept: it refers generally to the addition of the elements of an infinite set of real numbers, which are indexed by the points of some
lattice in N -dimensional space. A method of performing a lattice sum involves
accumulating the contributions of all these elements in some sequential order.
Unfortunately, the elements of the set are not, in general, absolutely summable so
the sequential order chosen can affect the answer. In this chapter we are concerned
with the particular lattice sums involved in the Madelung constant. Indeed, attaining specificity in the definition of the Madelung constant is our primary motive.
Although we are dealing with purely mathematical questions, it is our belief that
88
Convergence of lattice sums and Madelung’s constant
the results presented here may shed some light on the physics of crystals. Other
researchers [7, 8, 20] have expressed concern about the ambiguities involved in
summing a non-absolutely-convergent series in a different manner, but it appears
that no one has confronted this question fully.
Let L be a lattice in N -dimensional space and let A L = {al : l ∈ L} be a set
of real numbers indexed by L. There are two basic approaches to summing the
elements of A L : by direct summation or by integral transformations. The major
factors involved in choosing a method are physical meaningfulness and speed of
convergence.
The direct summation methods involve an orderly grouping of the elements of
A L into sequentially indexed finite subsets increasing in size to eventually include
any element of A L . Sometimes fractions of elements are included in the subsets
to maintain a physical principle such as electrical neutrality. Two commonly used
direct summation methods are due to Evjen [16] and Højendahl [23].
The most commonly used integral transformation method is known as the
Ewald method [17]. More recently the Mellin transformation applied to theta
functions has been used to put the integral transformation methods in a general context. An excellent review article by Glasser and Zucker [20] gives a
development of these methods and an extensive bibliography.
In this chapter we deal primarily with NaCl-type ionic crystals in two or three
dimensions. This is for two main reasons, the ease of notation in these cases and
the fact that almost every textbook introduces the Madelung constant on this type
of crystal first. From a mathematical and physical point of view there are two
very reasonable and simple direct summation methods that could be applied to
give the Madelung constant for an NaCl-type ionic crystal. One could take a basic
cube centred at the referenced ion with sides parallel to the basic vectors and
let the cube expand as the contributions from all lattice points within the cube
are accumulated. Alternately one could use expanding spheres centred at the reference ion. This latter method is intuitively appealing since all ions that are an
equal distance from the reference ion are given equal treatment. Thus, many textbooks [e.g., 1, 26]
some research
articles) write down the resulting infinite
√
√ (and √
series (6 − 12/ 2 + 8/ 3 − 6/ 4 + · · · ) as giving the Madelung constant for
an NaCl-type ionic crystal. Unfortunately, this infinite series does not converge.
This was proven by Emersleben [15] and, in light of the fact that most people are
unaware of this divergence, we include a short elementary proof in Theorem 2.3.
Section 2.2 is devoted to the two-dimensional square lattice while Section 2.3
contains the above-mentioned result on expanding spheres. In Theorem 2.4
we prove that the method of expanding cubes converges. In Section 2.4, the
mathematical tools become more sophisticated as we consider integral transformation methods and their relation to the direct summation methods dealt with in
Section 2.3. We have included, in Section 2.5, a careful analysis of some direct
summation methods in two dimensions in light of their property of being analytic
in the inverse power exponent. This analysis is quite illustrative of the relations
2.2 Two dimensions
89
between the various summation methods. In Section 2.6 we do a brief analysis of
a two-dimensional hexagonal lattice. Section 2.7 gives our conclusions.
2.2 Two dimensions
It is convenient to introduce the notation in the two-dimensional case of a simple
lattice in the plane with unit charges located at integer lattice points ( j, k) and
of sign (−1) j+k . The potential energy at the origin due to the charge at ( j, k) is
−(−1) j+k /( j 2 + k 2 )1/2 . If we want the total potential energy at the origin due to
all other charges, then we must sum all the numbers in the following set:
+
*
A = (−1) j+k /( j 2 + k 2 )1/2 : ( j, k) ∈ Z/(0, 0) ,
where Z denotes the integers. Because the elements of the subset of A with j = k
form a set of positive numbers with divergent sum, it is clear that the value of
the sum is highly dependent on the order in which the elements of A are added.
It is not immediately clear that any reasonable method will produce a convergent series. In addition, for the model to be physically relevant, all ‘reasonable’
methods should converge to the same number.
Here are two very reasonable methods.
First, consider the total potential due to all the charges within a circle of radius
r about the origin and let r → ∞. This leads to the series
∞
(−1)n C2 (n)
n 1/2
n=1
,
(2.2.1)
where C2 (n) is the number of ways of writing n as the sum of two squares of
integers (positive, negative, or zero). In deriving (2.2.1), we use the fact that
2
2
(−1) j+k = (−1) j +k = (−1)n , for any j, k ∈ Z with j 2 + k 2 = n. We will
refer to (2.2.1) as the method of expanding circles.
Second, there is the method of expanding squares. This is intuitively appealing,
as a perfect crystal grows by expansion of the shape of a basic unit cell. For each
natural number n, let
(−1) j+k
:
−n
≤
j,
k
≤
n
and
(
j,
k)
=
(0,
0)
.
(2.2.2)
S2 (n) =
( j 2 + k 2 )1/2
Then
lim S2 (n)
n→∞
is a way of expressing the series obtained by expanding squares.
It turns out that both these methods converge, as we will now show.
Theorem 2.1
The series in (2.2.1) converges.
Proof To carry out the proof that the series in (2.2.1) converges we introduce
some notation and standard facts from number theory. For any sequence of real
90
Convergence of lattice sums and Madelung’s constant
numbers {an }∞
β real we write an = O(n β ) if the sequence {n −β an } is
n=1 and
n
bounded. Let An = k=1 C2 (k), for each natural number n. Then An denotes the
number of non-origin lattice points inside or on a circle of radius n 1/2 . It is fairly
easy to see that An should be approximately π n; in fact, the reader can easily
show that An − π n = O(n 1/2 ). However, this is not quite good enough for us
here so we quote a stronger result, which can be found in Dickson [14]:
An − π n = O(n α )
for some α,
1
4
< α < 13 .
(2.2.3)
For a natural number n, a divisor of n is a natural number d such that d divides n.
Let d(n) denote the number of divisors of n and let dk (n) denote the number of
divisors d of n with d congruent to k modulo 4, for k = 1 or 3. With this notation,
Theorem 278 of Hardy and Wright [22] implies that
C2 (n) = 4[d1 (n) − d3 (n)].
(2.2.4)
This together with Theorem 315 of Hardy and Wright [22] gives
C2 (n) = O(n δ )
for any δ > 0.
(2.2.5)
Note that dk (2n) = dk (n) for k = 1 or 3 and any n. So
C2 (2n) = C2 (n)
Let Bn =
n
k=1 (−1)
kC
2 (k).
B2n =
for all natural numbers n.
(2.2.6)
Then
2n
(−1)k C2 (k)
k=1
=
n
C2 (2k) −
k=1
=2
n
C2 (2k − 1)
k=1
n
k=1
C2 (2k) −
2n
C2 (k).
k=1
Using (2.2.6),
B2n = 2An − A2n .
From (2.2.3) and (2.2.7), we have that, with α as in (2.2.3),
B2n = O(n α ).
Furthermore, this along with (2.2.5) implies that
B2n+1 = B2n − C2 (2n + 1) = O(n α ).
(2.2.7)
2.2 Two dimensions
91
Therefore
Bn = O(n α ).
(2.2.8)
Now consider the partial sums of the series in (2.2.1):
Tn =
n
(−1)k C2 (k)
k 1/2
k=1
= O(n α−1/2 ) −
Bn
=
+
Bk k −1/2 − (k + 1)−1/2
(n + 1)1/2
n
k=1
n
Bk (k + 1)−1/2 − k −1/2 .
(2.2.9)
k=1
By the mean value theorem, [(k + 1)−1/2 − k −1/2 ] ≤ 12 k −3/2 and therefore
Bk (k + 1)−1/2 − k −1/2 = O(k α−3/2 ).
[(k + 1)−1/2 − k −1/2 ] converges absolutely. Since
Since α − 32 < −1, ∞
k=1 Bk 1
−1/2 − k −1/2 ] exists. That is, the
α − 2 < 0, limn→∞ Tn = − ∞
k=1 Bk [(k + 1)
series in (2.2.1) converges.
We now turn to the limit in (2.2.2). We need an easy lemma from calculus that
will be left to the reader to verify. This lemma will also be used in the proof of
Theorem 2.4.
Lemma 2.1 For any positive real numbers a, b, and s, each of the following
functions is monotonically decreasing in t, for 0 < t < ∞:
f 1,s (t) = t −s ,
f 2,s (t) = t −s − (t + a)−s ,
f 3,s (t) = t −s − (t + a)−s − (t + b)−s + (t + a + b)−s .
Theorem 2.2
The limit in (2.2.2) exists.
Proof We apply the lemma to f 2,s with a = (k + 1)2 − k 2 and s = 12 . Then, if
j ≥ 0 and k ≥ 0 with j + k ≥ 1, we have
f 2,1/2 ( j 2 + k 2 ) > f 2,1/2 (( j + 1)2 + k 2 ).
Explicitly, we have
( j 2 + k 2 )−1/2 − [ j 2 + (k + 1)2 ]−1/2 − [( j + 1)2 + k 2 ]−1/2
+ [( j + 1)2 + (k + 1)2 ]−1/2 > 0.
(2.2.10)
Let g( j, k) denote the left-hand side of (2.2.10). Then (−1) j+k g( j, k) is the contribution to the potential at the origin due to a basic cell of four adjacent ions with
the closest ion at ( j, k). Inequality (2.2.10) says that the contribution always has
the same sign as that of the nearest ion.
92
Convergence of lattice sums and Madelung’s constant
Rewrite S2 (n) using the symmetries to get
S2 (n) = 4Q(n) + 4X (n),
where
Q(n) =
X (n) =
n
j,k=1
n
k=1
(−1) j+k
,
( j 2 + k 2 )1/2
(−1)k
.
k
Since limn→∞ X (n) = − ln 2, if we prove that limn→∞ Q(n) exists then the limit
in (2.2.2) will exist. We will establish a number of properties of the sequence
{Q(n)}∞
n=1 , which will be used to prove its convergence.
Property 1:
Q(2n) − Q(2n − 2) > 0
for all n ≥ 2.
That is, the even indexed elements increase. To see this we group the terms of
Q(2n) − Q(2n − 2) into basic cells of four, as illustrated in Fig. 2.1a for Q(6) −
Q(4). Thus,
Q(2n) − Q(2n − 2) =
n
g(2l − 1, 2n − 1) +
l=1
n−1
g(2n − 1, 2m − 1),
m=1
where as before, g( j, k) denotes the left-hand side of (2.2.10). So property
1 holds.
Property 2:
Q(2n + 1) − Q(2n − 1) < 0 for all n ≥ 1.
7
7
6
6
5
5
4
4
3
3
2
2
1
1
1
2
3
4
(a)
5
6
7
1
2
3
4
5
(b)
Figure 2.1 Illustrations of (a) property 1 and (b) property 2.
6
7
2.3 Three dimensions
93
That is, the odd-indexed elements of the sequence decrease. Referring to Fig. 2.1b
and correcting for the overlap at the (2n, 2n) point we are led to the following
grouping:
Q(2n + 1) − Q(2n − 1) =
n
n
[−g(2l − 1, 2n)] +
[−g(2n, 2m − 1)]
l=1
m=1
1
1
− √ −
√ < 0.
n 2 (n + 1) 2
Property 3:
Q(2n + 1) − Q(2n) > 0 for all n ≥ 1.
Thus, the odd-indexed elements are all greater than any even-indexed element.
This is clear again from a simple grouping of terms and using the monotonicity
of f 1,1/2 from the lemma:
n 1
Q(2n + 1) − Q(2n) = 2
((2l − 1)2 + (2n + 1)2 )1/2
l=1
1
1
+
−
√ > 0.
(4l 2 + (2n + 1)2 )1/2
(2n + 1) 2
Property 4:
lim Q(2n + 1) − Q(2n) = 0.
n→∞
Thus, the difference between successive elements goes to zero. To see this, simply
note that
1
2
+
0 < Q(2n +1)− Q(2n) <
√ → 0 as n → ∞.
[1 + (2n + 1)2 ]1/2 (2n + 1) 2
It is now easy to see that properties 1–4 imply that limn→∞ Q(n) exists.
This completes the proof of Theorem 2.2. Thus, we have shown that two
of the most obvious methods of summing for a Madelung constant in two
dimensions each converge. At this point, no indication has been given that the
two methods yield the same number. That this is indeed so will be shown in
Section 2.5.
2.3 Three dimensions
In this section the three-dimensional case will be considered. For the Madelung
constant of an NaCl-type crystal one must investigate ways of summing the
elements of the following set:
+
*
B = (−1) j+k+l /( j 2 + k 2 + l 2 )1/2 : ( j, k, l) ∈ Z3 /(0, 0, 0) .
94
Convergence of lattice sums and Madelung’s constant
In analogy with the two-dimensional case we will consider the method of
expanding spheres about the origin and the method of expanding cubes.
Our next theorem is a negative result, which is quite startling. Many textbooks
in physical chemistry and solid state physics give the series dealt with in Theorem
2.3 as the Madelung constant for an NaCl-type crystal [1, 26]. It also appears
in research articles and undergraduate courses. Although no one sums this series
directly, it is physically misleading to believe that it converges to anything.
Let C3 (n) denote the number of ways of writing n as a sum of three squares. If
we consider a sphere centred at the origin in three-space, add all the elements of
B that correspond to lattice points within the sphere, and then let the radius go to
infinity, we are led to the infinite series
∞
(−1)n C3 (n)
.
√
n
(2.3.1)
n=1
Theorem 2.3 (Emersleben [15]) The series in (2.3.1) diverges.
Proof It is interesting that the proof that the series in (2.3.1) diverges is much less
sophisticated than the proof in Theorem 2.1 that the series in (2.2.1) converges.
Our main tool is a simple estimate of the number of non-zero lattice points on
√
or
√ inside a sphere of radius r . Call this number L r . Notice that, for n ≤ r <
n + 1,
n
C3 (k).
Lr =
k=1
We leave to the reader the easy task of verifying that
4
L r − πr 3 = O(r 2 ).
3
This implies that
Lr
4
= π.
3
r3
Proceeding with a proof by contradiction we assume that
lim
r →∞
(2.3.2)
∞
(−1)n C3 (n)
converges.
√
n
n=1
√
This implies that n = C3 (n)/ n → 0 as n → ∞. For a natural number N , let
M N = max{n : n ≥ N }. Then M N → 0 as N → ∞. Fix N for the moment and
consider, for n > N , the inequality
n
N
n
√
√
√
L √n
k k ≤ n −3/2
k k + M N n −3/2
k .
√ 3 = n −3/2
( n)
k=1
k=1
k=N +1
(2.3.3)
2.3 Three dimensions
Now
n
√
k≤
k=N +1
n+1
N +1
t 1/2 dt =
2
3
95
(n + 1)3/2 − (N + 1)3/2 .
Inserting this in (2.3.3) implies that
N
√
L √n
n + 1 3/2
N + 1 3/2
−3/2
2
.
≤n
k k + 3 M N
−
√
n
n
( n)3
k=1
√
Letting n → ∞, we see that lim supn→∞ L √n /( n)3 ≤ 23 M N , for any N . Since
M N → 0 as N → ∞, we have that
L √n
√ 3 = 0.
n→∞ ( n)
lim
This is a contradiction of (2.3.2). Therefore,
∞
(−1)n C3 (n)
√
n
diverges.
n=1
In fact, the contributions of individual spherical shells do not tend to zero.
So it is not at all appropriate to define the Madelung constant via the method of
expanding spheres. We turn to the method of expanding cubes. Let
(−1) j+k+l
: −n ≤ j, k, l ≤ n, ( j, k, l) = (0, 0, 0) .
S3 (n) =
( j 2 + k 2 + l 2 )1/2
Theorem 2.4
The limit limn→∞ S3 (n) exists.
Proof We proceed as in the proof of Theorem 2.2. For j, k, l ≥ 1 let
g( j, k, l)
= ( j 2 + k 2 + l 2 )−1/2 − [( j + 1)2 + (k + 1)2 + (l + 1)2 ]−1/2
+ [( j + 1)2 + (k + 1)2 + l 2 )−1/2
+ [( j + 1)2 + k 2 + (l + 1)2 ]−1/2 + [ j 2 + (k + 1)2 + (l + 1)2 ]−1/2
− [( j + 1)2 + k 2 + l 2 ]−1/2 − [ j 2 + (k + 1)2 + l 2 ]−1/2
− [ j 2 + k 2 + (l + 1)2 ]−1/2 .
Then (−1) j+k+l g( j, k, l) represents the contribution to the potential at the origin of a basic unit cell whose closest corner is at ( j, k, l). An appropriate use
of the monotonicity of f 3,1/2 from Lemma 2.1 shows that g( j, k, l) > 0 for all
j, k, l ≥ 1.
96
Convergence of lattice sums and Madelung’s constant
Let
h(k, l) = (2k 2 + l 2 )−1/2 − [2k 2 + (l + 1)2 ]−1/2
− [2(k + 1)2 + l 2 ]−1/2 + [2(k + 1)2 + (l + 1)2 ]−1/2 .
Using f 2,1/2 , we get that h(k, l) > 0 for all k, l ≥ 1.
Let P(n) denote the part of S3 (n) that comes from the positive octant. That is,
for n ≥ 1,
n
(−1) j+k+l
.
P(n) =
( j 2 + k 2 + l 2 )1/2
j,k,l=1
Then, in a manner similar to that used for the Q(n), it can be shown that
limn→∞ P(n) exists. We proceed with the details of this demonstration.
The following identities are most easily seen by drawing a three-dimensional
version of Fig. 2.1, but they can be verified directly:
P(2n + 1) − P(2n − 1) = 3
n
n
%
&
g 2n, 2k − 1, 2l − 1 + 3
h(2n, 2 j − 1)
k,l=1
j=1
1
1
1
+√
−
,
2n + 1
3 2n
n
P(2n + 2) − P(2n) = −3
g(2n + 1, 2k − 1, 2l − 1)
−3
(2.3.4)
k,l=1
n
g(2n + 1, 2n + 1, 2 j − 1)
j=1
− g(2n + 1, 2n + 1, 2n + 1).
(2.3.5)
Both (2.3.4) and (2.3.5) hold for all n ≥ 1. From (2.3.4) and (2.3.5) we get the
properties of odd- or even-element monotonicity of the sequence of P(n).
Property 1 :
P(2n) − P(2n − 2) < 0 for all n ≥ 2.
Property 2 :
P(2n + 1) − P(2n − 1) > 0
for all n ≥ 1.
Notice that the inequalities are reversed from those of properties 1 and 2 in the
two-dimensional case. To get the analogues of properties 3 and 4 for the P(n) we
need to refer to the lemma for a final time. For n, j, k ≥ 1, let
h 0 (n, j, k) = (n 2 + j 2 + k 2 )−1/2 − [n 2 + ( j + 1)2 + k 2 ]−1/2
− [n 2 + j 2 + (k + 1)2 ]−1/2 + [n 2 + ( j + 1)2 + (k + 1)2 ]−1/2 .
2.3 Three dimensions
97
With a = (k + 1)2 − k 2 ,
h 0 (n, j, k) = f 2,1/2 (n 2 + j 2 + k 2 ) − f 2,1/2 (n 2 + ( j + 1)2 + k 2 ),
which is positive for all n, j, k ≥ 1. With this notation,
P(2n + 1) − P(2n) = −3
−3
n
h 0 (2n + 1, 2 j − 1, 2k − 1)
j,k=1
n
*
[2(2n + 1)2 + (2l − 1)2 ]−1/2
l=1
− [2(2n + 1)2 + (2l)2 ]−1/2
−
1
√ .
(2n + 1) 3
+
(2.3.6)
This leads to the following property.
Property 3 :
P(2n + 1) − P(2n) < 0 for all n ≥ 1.
Therefore, the decreasing even-indexed elements are all greater than the increasing odd-indexed elements. To see that there is a unique limit to the sequence of
P(n), we only need the last property, which implies that the distance between
successive terms approaches zero. This follows from (2.3.6) and
P(2n + 1) − P(2n) > −
3
1
−
√ .
2
1/2
[(2n + 1) + 2]
(2n + 1) 3
(2.3.7)
To verify (2.3.7) let
xj =
2n
k=1
(−1)1+ j+k
[(2n + 1)2 + j 2 + k 2 ]1/2
for 1 ≤ j ≤ 2n + 1.
Using the function h 0 , defined above, write
|x j | − |x j+1 | =
n
h 0 (2n + 1, j, 2k − 1) > 0.
(2.3.8)
k=1
Note that x j itself is an alternating sum of decreasing terms, so the sign of x j is
(−1) j . With (2.3.8), this implies that
0>
2n+1
j=1
x j > x1 >
−1
.
[(2n + 1)2 + 2]1/2
98
Convergence of lattice sums and Madelung’s constant
Then
P(2n + 1) − P(2n) = 3
2n+1
xj −
j=1
3
√ >−
[(2n + 1)2 + 2]1/2
(2n + 1) 3
1
1
√ .
(2n + 1) 3
−
So (2.3.7) holds. Thus, the following property has been established.
Property 4 :
lim P(2n + 1) − P(2n) = 0.
n→∞
Properties 1 –4 imply that limn→∞ P(n) exists. Finally,
S3 (n) = 8P(n) + 12Q(n) + 6X (n),
(2.3.9)
where, as before,
Q(n) =
n
j,k=1
(−1) j+k
( j 2 + k 2 )1/2
and
X (n) =
n
(−1)k
k=1
k
.
Since each term on the right-hand side of (2.3.9) approaches a limit as n → ∞,
we have that limn→∞ S3 (n) exists.
Remark 1: Although this method of summing over expanding cubes is not rapidly
convergent, it is extremely well behaved. The alternations of P(n) and Q(n)
above and below their limiting values provide precise error bounds, which may
be useful in theoretical considerations.
Remark 2: The work of Campbell [8] must be mentioned at this point. He states
general conditions on a doubly indexed series and concludes a convergence result,
which is stronger than Theorem 2.2 above. However, there is a serious error in his
proof and his general theorem is false. A simplified version of Campbell’s claimed
result would be the following. Let {ai j }i∞= 1 ,∞
j = 1 be a doubly indexed ‘sequence’
of reals satisfying: (i) for all i, {|ai1 |, |ai2 |, |ai3 |, . . .} is a monotonically decreasing sequence with lim j→∞ ai j = 0 and, for all j, {|a1 j |, |a2 j |, |a3 j |, . . .} is a
monotonically decreasing sequence with limi→∞ ai j = 0; (ii) the sign of ai j is
∞ ∞
( j=1 ai j ) exists.
(−1)i+ j+1 . Then i=1
Here is a counterexample to this claim. Let the ai j be defined as in the array in
Table 2.1. Let
∞
ai j for i = 1, 2, 3, . . .
Ui =
j=1
2.3 Three dimensions
99
Table 2.1 An array ai j that forms a counterexample to the general
convergence result claimed by Campbell
i\ j
1
2
3
4
5
6
.
.
.
1
2
1
− 12 − 10−2
− 21
1 − 10−3
2
− 31
1 − 10−4
3
− 41
1 − 10−5
4
1
2
1
− 3 − 10−3
1
3
− 14 − 10−4
3
1
3
− 14 − 10−4
1
4
1
− 5 − 10−5
1
5
− 16 − 10−6
.
.
.
.
.
.
.
.
.
4
5
− 41
1 − 10−5
4
− 51
1 − 10−6
5
− 61
1 − 10−7
6
.
.
.
1
5
− 16 − 10−6
1
6
1
− 7 − 10−7
1
7
− 18 − 10−8
.
.
.
···
···
···
···
···
···
Clearly, each Ui exists and the sign of Ui is (−1)i+1 . The odd-indexed Ui are all
positive and easily calculated:
U1 = ln 2 = 1/(1 × 2) + 1/(3 × 4) + 1/(5 × 6) + · · · ,
U3 = 1 − ln 2 = 1/(2 × 3) + 1/(4 × 5) + 1/(6 × 7) + · · · .
In general,
*
U2k−1 = (−1)k+1 ln 2 − 1 −
=
∞
j=0
1
2
+ · · · + (−1)k /(k − 1)
1
,
(k + 2 j)(k + 2 j + 1)
k ≥ 2.
Any U2k is negative and is given by
∞
U2k =
j=k+1
−
1
10−k
.
=−
j
10
9
Note that on the one hand
∞
U2k = −
k=1
∞
1
1 −k
=− .
10
9
81
k=1
On the other hand,
∞
U2k−1 =
k=1
∞ k=1
1
1
+
+ ···
k(k + 1) (k + 2)(k + 3)
∞ ∞ ∞
1
1 ∞ 1
>
>
dt
4
4 j2
t2
n
n=1 j=n
=
1
4
∞
n=1
1
= ∞.
n
n=1
+
100
Convergence of lattice sums and Madelung’s constant
Since the sum of the positive terms diverges and the sum of the negative terms
n
Ui does not exist. Thus
converges, it follows that limn→∞ i=1
∞
∞
diverges,
ai j
i=1
j=1
even though it satisfies Campbell’s conditions for convergence. Campbell goes on
to claim that the analogous result holds for any dimension and that one could also
prove convergence if one summed by expanding rectangles. Both these statements
are unfounded.
Remark 3: In light of the above, it appears that there is no simple proof in the literature of the convergence of any of the most elementary direct summation methods.
That is why detailed proofs of Theorems 2.2 and 2.4 are given. Emersleben’s
result (Theorem 2.3) indicates that a non-casual approach is justified.
Remark 4: The proofs given of Theorems 2.2 and 2.4 are simple and intuitive,
based as they are on the fact that the contribution of a basic unit cell to the sum
always has the same sign as that of the nearest point in the cell to the origin. We
have abstracted this property and have obtained quite general convergence results
for multidimensional alternating series (see for instance Chapter 8). We will point
out later that Theorems 2.2 and 2.4 also follow from the deeper considerations of
the next section.
2.4 Integral transformations and analyticity
Our purpose in this section is to establish a firm connection between the elementary direct summation methods discussed above and integral transformation
methods, which are described by Glasser and Zucker in their survey article [20].
One major consequence of this connection is that we can give a definition of the
Madelung constant, which has a firm mathematical foundation and is unique in
a strong enough sense to indicate why diverse methods of performing the lattice
sums lead to the same number. We begin with a general discussion of the analyticity of certain lattice sums in N -dimensional space. Of course N = 2 and 3 are
the most interesting cases, but the general notation is just as convenient.
For a complex number s, let Re s denote the real part of s and let
(−1)n
N
:
n
∈
Z
/{0}
,
A N (s) =
||n||2s
where n = (n 1 , n 2 , . . . , n N ) ∈ Z N , (−1)n = (−1)n 1 +n 2 +···+n N , and ||n|| =
(n 21 + n 22 + · · · + n 2N )1/2 . We also use the notations
%
&
|n| = |n 1 |, |n 2 |, . . . , |n N | ∈ Z N
2.4 Integral transformations and analyticity
101
and, for m ∈ Z N ,
if n j ≥ m j for 1 ≤ j ≤ N .
If Re s > N /2 then a simple comparison test shows that n=0 1/||n||2s < ∞.
So the elements of A N (s) are absolutely summable if Re s > N /2. Let
(−1)n
n
d N (2s) =
:
n
∈
Z
/{0}
.
||n||2s
n≥m
Then d N (z) is a function of the complex variable z for Re z > N . In fact, it is
a multidimensional zeta function, analytic on this domain. To see this define, for
m ∈ Z N , with m > 0,
(−1)n
n
:
n
∈
Z
/{0}
and
|n|
≤
m
.
(2.4.1)
d m (z) =
||n||z
Then d m (z) is analytic for Re z > 0. For fixed δ > 0, if Re z > N + δ,
1
N
N
|d N (z) − d m (z)| ≤
:
n
∈
Z
/{l
∈
Z
:
|l|
≤
m}
.
||n|| N +δ
(2.4.2)
The right-hand side of (2.4.2) can be made arbitrarily small by letting the minimal
coefficient of m become large. Thus, on the region (Re z > N + δ), d N (z) is
the uniform limit of analytic functions and is therefore analytic. Since δ > 0 is
arbitrary we have established the following proposition.
Proposition 2.1
The function d N (2s) is analytic in s for Re s > N /2.
Now comes the crucial step for the definition of the Madelung constant. The
functions d N (2s) can be analytically continued to the region (Re s > 0). To
accomplish this we follow the ideas of Glasser and Zucker [20] and introduce
θ -functions and the Mellin transform. We need, in particular,
θ4 (q) =
∞
2
(−1)n q n ,
0 ≤ q < 1.
n=−∞
Hence
θ4 (e−t ) =
∞
(−1)n e−n t ,
2
0 < t < ∞.
(2.4.3)
n=−∞
For a continuous function f (t) defined for 0 < t < ∞, bounded as t → 0,
and decaying sufficiently fast as t → ∞, one can define a normalized Mellin
transform Ms ( f ) for Re s > 0 by
∞
Ms ( f ) = −1 (s)
f (t)t s−1 dt,
0
102
Convergence of lattice sums and Madelung’s constant
where is the usual gamma function, given by
∞
(s) =
e−t t s−1 dt for Re s > 0.
0
−1
(that is, 1/ ) are analytic functions on (Re s > 0). A useful
Of course, and
property of the Mellin transform is that, for a > 0 and f such that its Mellin
transform exists, Ms (τa f ) = Ms ( f )/a s , where τa f (t) = f (at) for all t > 0. In
particular,
Ms (e−at ) = 1/a s
for a > 0.
(2.4.4)
Consider now a truncation of the series for θ4 . For some positive integer m, let
n n2
N
φm (q) = m
n=−m (−1) q . If m ∈ N , say m = (m 1 , . . . , m N ), then let m =
min{m 1 , . . . , m N }. We wish to approximate the N th power of θ4 with products of
φm i . For 0 < t < ∞,1
N
N −t
N
−t θ (e ) −
|θ4 (e−t ) − φm i (e−t )| + φm i (e−t )
φ
(e
)
mi
4
≤
i=1
i=1
−
N
N
N
φm i (e−t ) =
(bi + ai ) −
ai ,
i=1
i=1
i=1
(2.4.5)
where ai = φm i (e−t ) and bi = |θ4 (e−t ) − φm i (e−t )|. Note that 0 < ai , bi < 1
N
N
(bi + ai ) − i=1
ai ,
for i = 1, . . . , N , and the last expression in (2.4.5), i=1
represents the difference in volume between an N -box of side lengths bi + ai , i =
N
(bi + ai ) −
1, . . . , N and one of side lengths ai , i = 1, . . . , N . Clearly i=1
N
N −1 . Now |θ (e−t ) − φ (e−t )| is the
a
<
max{b
:
1
≤
i
≤
N
}N
2
i
i
4
m
i=1
2
maximum bi and |θ4 (e−t ) − φm (e−t )| < 2e−m t . So (2.4.5) becomes
N
N −t
−t N −m 2 t
θ (e ) −
φ
(e
)
.
(2.4.6)
mi
4
< N2 e
i=1
N
We also need the Mellin transform of i=1
φm i (e−t ) − 1, which is easily found
using linearity and (2.4.4):
N
2
φm i (e−t ) − 1 = Ms
(−1)n e−||n|| t
Ms
|n|≤m
i=1
=
|n|≤m
(−1)n
||n||2s
= d (2s).
m
(2.4.7)
1 The derivation following, up to the end of (2.4.6), contains minor mathematical errors and does
not provide sharp estimates; subsequent appearances of N 2 N in Theorem 2.6 are also not sharp.
These issues are addressed in the Commentary at the end of the chapter: see (2.8.1).
2.4 Integral transformations and analyticity
103
The prime on the summation sign indicates that the n = 0 term is omitted. We are
now ready for the main theorems of this section. Define F(s) = Ms [θ4N (e−t ) − 1]
wherever it exists.
Theorem 2.5 The Mellin transform F(s) of θ4N (e−t ) − 1 exists and is analytic
for all s with Re s > 0. Furthermore F provides an analytic continuation of
d N (2s) to the region (Re s > 0).
Theorem 2.6
Re s > 0,
For any m ∈ Z N , m > 0, m = min{m i : 1 ≤ i ≤ N }, and
|F(s) − d m (2s)| < N 2 N (Re s)/(m 2 Re s |(s)|).
Proof of Theorems 2.5 and 2.6 combined
Re s > 0. Since
Let s be a complex number such that
0 ≤ 1 − θ4N (e−t ) ≤ N [1 − θ4 (e−t )] ≤ N e−t
then
θ N (e−t ) − 1 |t s−1 | dt ≤ N
4
∞
0
Therefore, if Re s > 0 then
∞
0
(2.4.8)
∞
for all 0 < t < ∞,
e−t t Re s−1 dt = N (Re s).
0
θ4N (e−t ) − 1 t s−1 dt = F(s)
exists. Using (2.4.6) and (2.4.7), with m as in Theorem 2.6,
∞
N
N −t
m
−t s−1
φm i (e ) |t | dt
|(s)| |F(s) − d (2s)| ≤
θ4 (e ) −
0
i=1
∞
≤ N 2N
e
−m 2 t
t Re s−1 dt
0
= N 2 N (Re s)/m 2 Re s .
Thus (2.4.8) holds. In turn, (2.4.8) implies that F(s) can be uniformly approximated by d m (2s) on any region of the form Rδ,M = {s : |s| < M and Re
s > δ}. To see this, let K be an upper bound for the continuous function
N 2 N (Re s)/|(s)| on the closure of Rδ,M . Then, for any > 0 and any m such
that m = min{m 1 , . . . , m N } > (K /)1/(2δ) ,
|F(s) − d m (2s)| < for all s ∈ Rδ,M .
Since is arbitrary and d m is analytic, F is analytic on Rδ,M for any δ > 0 and
0 < M < ∞. Therefore F is analytic on (Re s > 0). Finally, it is now clear
that F(s) agrees with d N (2s) if Re s > N /2. Thus F is an analytic continuation
of d N .
104
Convergence of lattice sums and Madelung’s constant
In light of Theorem 2.5 we will drop the use of F and write
d N (2s) = Ms [θ4N (e−t ) − 1]
for Re s > 0.
(2.4.9)
A rigorous mathematical definition can now be given for the Madelung
constant.
Definition 2.2 For a three-dimensional NaCl-type ionic crystal, the Madelung
constant is the number
d3 (1) = M1/2 [θ43 (e−t ) − 1].
Of course, this is the very number that has been approximated by many different methods over the years. We have just given a definition that avoids all the
ambiguities of meaning that have existed. The uniqueness of analytic continuation
explains the special significance of this particular sum of the elements of
+
*
(−1)i+ j+k /(i 2 + j 2 + k 2 )1/2 : (i, j, k) ∈ Z3 /(0, 0, 0) .
Formula (2.4.8) emphasizes the strong connection between integral transformation methods and direct summation methods. In fact it is worthwhile to formulate
a corollary to Theorem 2.6, which gives explicit error bounds for a finite sum
approximation to the Madelung constant.
Corollary 2.1
Let m i > 0 for i = 1, 2, 3 and m = min{m 1 , m 2 , m 3 }. Then
12
(−1)i+ j+k
d3 (1) −
<
.
2
2
2
1/2
(|i|,| j|,|k|)<m (i + j + k )
m
Remark 5: The above corollary says that the Madelung constant for NaCl can be
obtained not only by expanding cubes but by expanding any rectilinear shape, and
the order of convergence is the inverse of the minimum dimension. In fact, it is
permissible to let some coordinates go to infinity before others.
Remark 6: Of course Theorems 2.2 and 2.4 follow immediately from
Theorems 2.5 and 2.6 but we preferred to present the simple direct proofs from
Sections 2.2 and 2.3 for the reasons given in Remark 4.
2.5 Back to two dimensions
In this section we consider the analyticity of various methods of summing the
elements of the set
As = {(−1) j+k /( j 2 + k 2 )s : ( j, k) ∈ Z/(0, 0)}.
2.5 Back to two dimensions
105
From Theorem 2.6 it follows that the method of expanding squares leads to
d2 (2s), which is analytic for Re s > 0. In fact, expanding rectangles of any shape
with sides parallel to the axes leads to d2 (2s). In Theorem 2.1 we showed that
the method of expanding circles converged when s = 12 , but there is no reason
to believe that d2 (1) is obtained unless one shows that the appropriate function is
analytic. Using the notation of Section 2.2, let
G(s) =
∞
(−1)n C2 (n)
ns
n=1
,
(2.5.1)
whenever the right-hand side converges. Then G(s) is the sum of the elements of
As obtained by expanding circles.
Theorem 2.7 The function G(s) exists and is analytic for Re s > 13 . Thus,
n
1/2 .
G(s) = d2 (2s) if Re s > 13 ; in particular, d2 (1) = ∞
n=1 (−1) C 2 (n)/n
Proof As in the proof of Theorem 2.1, let Bn =
Bn = O(n 1/3− )
n
k=1 (−1)
kC
2 (k).
By (2.2.8),
for some > 0.
(2.5.2)
Define G n (s) for all s with Re s > 0 by
G n (s) =
n
(−1)k C2 (k)
ks
k=1
.
As in (2.2.9),
Bn
+
Bk k −s − (k + 1)−s .
(n + 1)s
n
G n (s) =
(2.5.3)
k=1
By (2.5.2), if Re s ≥
1
3
then |Bn /(n + 1)s | → 0, uniformly in s. Note that
|k −s − (k + 1)−s | = (−s)
k+1
k
k+1
≤ |s|
t −(s+1) dt t −(Re s+1) dt ≤ |s|k −(Re s+1) .
k
(2.5.4)
So, for s ∈ R M = {z : Re z ≥ 13 , |z| ≤ M} with M a fixed positive number and
1 ≤ N ≤ N ,
106
Convergence of lattice sums and Madelung’s constant
N
Bk k −s − (k + 1)−s
N
≤
|Bk | |k −s − (k + 1)−s |
k=N
k=N
N
≤K
k 1/3− |k −s − (k + 1)−s |
by (2.5.2)
k=N
≤ KM
N
k 1/3−−Re s−1
by (2.5.4)
k=N
1/3−−Re s
≤ KMN
≤ KMN −
→0
as N , N → ∞,
uniformly for s ∈ Rm . Thus the sequence of functions {G n (s)}∞
n=1 is uniformly
Cauchy on R M and it converges uniformly to a limit function G(s). Furthermore,
each G n (s) is analytic, so G(s) is analytic for s ∈ R M . Since M is arbitrary, G(s)
exists and is analytic for all s with Re s > 13 .
Remark 7: It is not known what the minimum non-negative β is, such that G(s)
exists for all s with Re s > β (see Chapter 8). However, if we consider another
method of summing the elements of As , we can get a very complete and illuminating analysis. This is the method of expanding diamonds. For each k = 1, 2, 3, . . .
and complex s with Re s > 0, let
δk (s) =
k
(k − j)2 + j 2
−s
.
j=0
For each n = 1, 2, 3, . . ., let
n (s) = 4
n
(−1)k δk (s) − 4X n (2s),
(2.5.5)
k=1
where X n (s) = nk=1 (−1)k /k s . Note that n (s) counts the contributions within
k
s
the diamond |k| + | j| ≤ n. Now, limn→∞ X n (s) = ∞
k=1 (−1) /k = −α(s),
and α is known to be analytic for Re s > 0. Therefore, in order to determine
for which s the limit of n (s) exists and is analytic, it is sufficient to analyze
∞
k
k=1 (−1) δk (s). We begin by establishing a number of facts about the sequence
of δk .
Proposition 2.2
√
√
1
k
(a) We have limk→∞ δk ( 12 ) = 2 ln( 2+1). Thus ∞
k=1 (−1) δk ( 2 ) diverges.
1
(b) For real r ≥ 2 , δk−1 (r ) ≥ δk (r ), k = 2, 3, 4, . . .
1
k
(c) The sum ∞
k=1 (−1) δk (s) exists and is analytic for Re s > 2 .
2.5 Back to two dimensions
107
Proof For (a),
δk ( 12 )
=
k
(k − j)2 + j 2
−1/2
j=0
=
k
1
j=0
k
1−
j
k
2
+
1
(1 − t)2 + t 2
→
0
√
√
= 2 ln( 2 + 1).
2 −1/2
j
k
−1/2
dt
as k → ∞
For the proofs of (b) and (c) it is very convenient to introduce the following
function. For Re s ≥ 12 , let
π/4
s
cos2s θ dθ − 1.
V (s) = 2s(2 )
0
V ( 12 )
= 0. If r ≥
Then V is continuous and
π/4
(2 cos2 θ )r dθ − 1 ≥
V (r ) = 2r
0
1
2
then
π/4 √
We proceed now to the proof of (b). Let r ≥
δk−1 (r ) − δk (r ) =
2 cos θ dθ − 1 = 0.
1
2
and k ≥ 2. Then
k ,
[(k − j)2 + ( j − 1)2 ]−r − [(k − j)2 + j 2 ]−r − k −2r
j=1
=
k j=1
≥
k j=1
j
j−1
j
j−1
2r t dt
− k −2r
[(k − j)2 + t 2 ]r +1
2r t dt
− k −2r
[(k − t)2 + t 2 ]r +1
k
2r t dt
− k −2r
2 + t 2 ]r +1
[(k
−
t)
0
1
2r u du
−2r
=k
−
1
2
2 r +1
0 [(1 − u) + u ]
1/2
1
(v + 2 ) dv
1
−
1
v
=
u
−
= k −2r 2r
2
1
−1/2 2r +1 (v 2 + 4 )r +1
1/2
2r
dv
−1
= k −2r r +1
2
2
(v + 14 )r +1
0
π/4
−2r
2 r
2r
(2 cos θ ) dθ − 1
(tan θ = 2v)
=k
=
(2.5.6)
0
0
= k −2r V (r ) ≥ 0.
108
Convergence of lattice sums and Madelung’s constant
That is, δk−1 (r ) ≥ δk (r ), for r ≥ 12 , k = 2, 3, 4, . . .
To prove (c), let > 0 and M < ∞ be arbitrary. Let
+
*
R = z : Re z > 12 + and |z| < M .
For s ∈ R, let r = Re s. For k ≥ 2, we can write
δk−1 (s) − δk (s) =
k−1 ,
[(k − j)2 + ( j − 1)2 ]−s − [(k − j)2 + j 2 ]−s
j=1
+ (k − 1)−2s − 2k −2s
k−1 j
2st dt
=
+ (k − 1)−2s − 2k −2s .
2
2 s+1
j−1 [(k − j) + t ]
j=1
Thus
|δk−1 (s) − δk (s)| ≤
k−1 j−1
j=1
≤ 2M
j
2|s|t dt
+ 3(k − 1)−2r
[(k − j)2 + t 2 ]r +1
k−1 t dt
+ 3(k − 1)−2r
[(k − 1 − t)2 + t 2 ]r +1
j−1
j=1
j
k−1
t dt
+ 3(k − 1)−2r
[(k − 1 − t)2 + t 2 ]r +1
0
1
u du
−2r
2M
= (k − 1)
+3
2
2 r +1
0 [(1 − u) + u ]
= 2M
= (k − 1)−2r (M/r )V (r ) + M/r + 3 ≤ (k − 1)−2r C,
(2.5.7)
where C is the maximum of the continuous function (M/r )V (r ) + M/r + 3 for
n
1
k=1 [δ2k−1 (s) − δ2k (s)] is an analytic function
2 + ≤ r ≤ M. Now, for each n,
of s for Re s > 12 , and
∞
n
k
δ2k−1 (s) − δ2k (s)
(−1) δk (s) = lim −
n→∞
k=1
k=1
exists uniformly on R, by (2.5.7) and the Weierstrass M-test. Since > 0 and
M < ∞ are arbitrary, (c) has been established.
We can now describe the behaviour of the diamond sums.
Theorem 2.8
1, 2, . . ., let
For each complex number s with Re s > 0 and each n =
n (s) =
n
l=1
(−1)
l
| j|+|k|=l
%
j +k
2
&
2 −s
.
2.6 The hexagonal lattice
Then limn→∞ n (s) exists and is analytic for Re s >
fails to converge,
d2 (1) =
lim
109
1
2.
Although {n ( 12 )}∞
n=1
lim n (r ) .
(2.5.8)
r →1/2+ n→∞
Proof These claims all follow immediately from Proposition 2.2.
Remark 8: Further analysis along the lines of Proposition 2.2 shows that although
n
% 2
&
l
2 −1/2
j +k
(−1)
l=1
| j|+|k|=l
is divergent, it is Cesàro summable or Abel summable to d2 (1).
The diamond sums provide a nice illustration of how a method of summing the
elements of As can be analytic in s, for Re s large, but then with decreasing Re s
this analyticity fails at a specific point. With the diamond sums it happens to be at
1
2 ; with expanding squares or rectangles it is at 0. It is not clear where it fails for
expanding circles; it is at some point less than 13 . In three dimensions the method
of expanding spheres fails at some point greater than 12 .
2.6 The hexagonal lattice
As an illustration of what is obtained when one studies other crystal lattices in the
above manner, we include a brief summary of results on the Madelung constant of
a two-dimensional regular hexagonal lattice with ions of alternating unit charge.
In order to obtain a tractable expression for the terms appearing in the lattice
sum, choose a coordinate system with an angle of φ = π/3 between the positive
axes. Then an arbitrary site in the lattice has coordinates (n, m), with n and m
integers. A charge of +1, −1, or 0 is attached to that site in a regular fashion (see
Fig. 2.2). By considering the two parallelograms indicated in Fig. 2.2, one can see
that this charge may be expressed by
q(n, m) =
+
4*
−sin nθ sin [(m −1)θ ]+sin [(m +1)θ ] sin [(n +1)θ ] ,
3
θ=
2π
.
3
The distance of the point (n, m) from the origin is given by
(n, m) = (n + m/2)2 + 3(m/2) 1/2 .
The set of numbers to be summed is then
2s
*
+
Cs = q(n, m)/(n, m) : (n, m) ∈ Z2 /(0, 0)
for Re s > 0.
As before, the elements of Cs are absolutely summable for Re s > 1 and we
require an analytic continuation of their sum to a region which includes s = 12 .
Arguments like those used for the diamond sums will show that direct summation
110
Convergence of lattice sums and Madelung’s constant
Figure 2.2 The hexagonal lattice.
by expanding shells of hexagons will converge analytically for Re s > 12 and even
have a limit as s approaches 12 from the right. However, for precise calculation
purposes an analytic continuation via integral transform methods is far superior.
Let
q(n, m)
: (n, m) ∈ Z2 /(0, 0) ,
(2.6.1)
H2 (2s) =
|(n, m)|2s
for Re s > 1. Then H2 is an analytic function of s and the series converges absolutely. Substituting the expression for q(n, m) and using elementary trigonometric
identities yields
√
cos[(m − n)θ ]
3 sin[(m − n)θ ]
−
,
(2.6.2)
H2 (2s) =
3
|(n, m)|2s
|(n, m)|2s
indicates that the sum is over (n, m) ∈ Z2 /(0, 0). By symmetry conwhere
siderations, the second term in the right-hand side of (2.6.2) is zero. Further
manipulation of theta functions (using the modular equation of order 3) produces
the rectangular sum
1
1 (−1)n+m
. (2.6.3)
−
H2 (2s) = (31−s − 1)
2
(n 2 + 3m 2 )s
(n 2 + 3m 2 )s
A theta function identity due to Cauchy (see Dickson [14]) and a Mellin transform
yields
H2 (2s) = 3(31−s − 1)ζ (s)L −3 (s),
−s
where ζ (s) is the standard zeta function ( ∞
n=1 n ) and
L −3 (s) = 1 − 2−s + 4−s − 5−s + 7−s − 8−s + · · · .
(2.6.4)
2.7 Concluding remarks
111
The formula (2.6.4) can also be deduced directly from (2.6.2) by using results
in Section IV of Glasser and Zucker [20]. While the intermediate sums (2.6.3)
and (2.6.4) are only analytic for Re s > 1, the standard continuation of the zeta
function,
(1 − 21−s )ζ (s) =
∞
(−1)n+1 n −s = α(s),
n=1
gives
H2 (2s) = 3(31−s − 1)(1 − 21−s )−1 α(s)L −3 (s).
(2.6.5)
The right-hand side of (2.6.5) is an analytic function of s for Re s > 0 and therefore (2.6.5) provides the required analytic continuation, which is necessary for the
Madelung constant of this hexagonal crystal lattice:
√ % &
√
% &
(2.6.6)
H2 (1) = 3(1 − 3)(1 + 2)α 12 L −3 12 .
This can be considered as a solution to this lattice sum problem, as both α( 12 ) and
L −3 ( 12 ) can be rapidly calculated by known techniques. At s = 1, we have an
exact result:
π ln 3
(2.6.7)
H2 (2) = − √ .
3
2.7 Concluding remarks
We have investigated some fundamental properties of the multiply indexed series
involved in the definition of the Madelung constant for an NaCl-type ionic crystal
in two and three dimensions. We have provided elementary proofs that convergent
series are obtained if the series is summed by letting the shape of a basic unit cell
expand. The natural method of summing the effects of all ions within a fixed
distance and letting the distance go to infinity leads to a convergent series in two
dimensions but not in three dimensions.
We have provided a unity to the concept of the Madelung constant by the use of
the analytic continuation of a complex function. Thus, although the expression for
the Madelung constant is conditionally convergent when summed by expanding
squares (or cubes), other methods of summing will provide the same answer provided that they are ‘analytic’ in the correct sense. We have provided an analysis
of the methods of expanding circles and expanding diamonds in two dimensions
to illustrate this point.
Perhaps the most important results are those in Section 2.4, relating the integral transformation methods and the direct summation methods. These integral
transform methods are the most useful in practice as they lead to very rapidly
convergent series.
112
Convergence of lattice sums and Madelung’s constant
In the course of these investigations we have encountered many curious facts,
most of which are probably known to experts in the area. However, the formulas
(2.6.6) and (2.6.7) seem to be unknown and to be of sufficient interest to have been
included, at least as an illustration that the techniques of analytic continuation are
applicable to other lattices.
2.8 Commentary: Improved error estimates
Equation (2.4.5) and the subsequent proof of (2.4.6) contain errors resulting from
the incorrect manipulation of inequalities. Here we provide a correct proof which
simultaneously provides better error estimates for convergence.
Without loss of generality, let m = m 1 ≤ m 2 ≤ · · · ≤ m N . We wish to
N
φm i (e−t )| for 0 < t < ∞. For notational convenience,
bound |θ4N (e−t ) − i=1
N
−t
let θ := θ4 (e ) and let f i := φm i (e−t ). Observe first that 0 < θ, | f i | ≤ 1. Then
observe, that, for |a|, |b| ≤ 1, |a n −bn | ≤ n|a−b| by factorization and the triangle
inequality. By these observations and repeated use of the triangle inequality, we
have
N
θ − f 1 f 2 · · · f N ≤ θ N − f 1N + | f 1 | f 1N −1 − f 2 f 3 · · · f N ≤ N |θ − f 1 | + f 1N −1 − f 2N −1 + | f 2 | f 2N −2 − f 3 f 4 · · · f N ≤ N |θ − f 1 | + (N − 1)| f 1 − f 2 | + f 2N −2 − f 3N −2 + | f 3 | f 3N −3 − f 4 f 5 · · · f N ..
.
≤ N |θ − f 1 | + (N − 1)| f 1 − f 2 | + (N − 2)| f 2 − f 3 | + · · · + | f N −1 − f N |.
Now, as θ and each f i is an alternating series, it is easy to see that each absolute
2
value is less than 2e−m t . Summing the arithmetic progression above, we have
established that
N
2
N −t
φm i (e−t ) < N (N + 1)e−m t .
(2.8.1)
θ4 (e ) −
i=1
Equation (2.8.1) should be used in place of (2.4.6). In particular, in the statement
(and proof) of Theorem 2.6, N 2 N should be replaced by N (N + 1), which is a
tighter estimate for the error in the convergence.
2.9 Commentary: Restricted lattice sums
The theta function method is easily adapted to yield possibly useful expressions
for lattice sums subjected to various geometric restrictions. Here, this is illustrated
2.9 Commentary: Restricted lattice sums
113
by two examples of the form
F(R) =
1
,
Q(ρ)s
ρ
(2.9.1)
where ρ denotes a point of a crystal lattice in some dimension, lying in or on
a sphere of radius R centred at one of the lattice points, which may or may not
be included in the sum. The quantity Q represents a quadratic form, and s is
usually a real parameter; they play an essential role in many areas of chemistry
and physics [20] and in the mathematical study of the corresponding infinite sums
(see the discussion earlier in this chapter). We finish with a third example, where
the restriction is to a cone.
Example 2.1 A restricted Lorenz–Hardy sum is given by
F(R) =
(m,n)=(0,0)
(m 2
1
.
+ n 2 )s
(2.9.2)
For s = 12 this sum is required, for example, in studying the crystallization of
a low-density electron gas on a metal surface [3] and for s = 0 (2.9.2) presents
Gauss’ famous problem of counting lattice points enclosed in a circle [21] (in this
case, centred at the lattice point at the origin).
We proceed by means of a series of simple integral transformations. As usual,
the Heaviside step function is denoted by θ . Thus,
F(R) =
=
1
(s)
(m,n)=(0,0)
∞
0
dt
t 1−s
θ (R 2 − m 2 − n 2 )
(m 2 + n 2 )s
θ (R 2 − m 2 − n 2 )e−t (m
2 +n 2 )
.
(2.9.3)
(m,n)=(0,0)
Throughout, it is assumed that s has values for which such operations are justified.
Deficiencies later can usually be corrected by analytic continuation or judicious
subtractions. Next, the step function is replaced by its representation as an inverse
Laplace transform:
1
F(R) =
2πi(s)
c+i∞
c−i∞
dz R 2 z
e
z
0
∞
dt
t 1−s
(m,n)=(0,0)
e−(z+t)(m
2 +n 2 )
. (2.9.4)
114
Convergence of lattice sums and Madelung’s constant
The double summation is easily carried out in terms of the Jacobi theta function
[21], yielding
c+i∞
1
dz R 2 z ∞ dt 2
e
F(R) =
[θ (0, q) − 1]
2πi(s) c−i∞ z
t 1−s 3
0
∞
c+i∞
R2
∞ ∞
4
dt
dz uz =
du
(−1)m q n(2m+1) ,
e
(s) 0
t 1−s c−i∞ 2πi
0
n=1 m=0
(2.9.5)
where q = exp[−(z + t)] and the theta function has been re-expressed by means
of one of Jacobi’s famous q-identities [21],
(−1)m q n(2m+1) .
(2.9.6)
θ32 (0, q) = 1 + 4
m,n
The next step is to substitute y = z + t, which produces a shift in the positive constant c in the inverse Laplace transform. However, this can be undone by shifting
the contour back as allowed by Cauchy’s theorem, since no singularities intervene.
Therefore,
∞
R2
dt −ut c+i∞
4
du
e
dy euy
(−1)m e−yn(2m+1)
F(R) =
s−1
(s) 0
t
0
c−i∞
m,n
2
∞
∞
R
∞
dt −ut
4 =
(−1)m
du
e δ[u − n(2m + 1)], (2.9.7)
1−s
(s)
t
0
0
n=1 m=0
since the y-integral is a representation for the Dirac delta function. Performing
the integrals now simply reverses the steps followed, and we have
F(R) = 4
∞
(−1)
N
(R,m)
∞
− n(2m + 1)]
(−1)m
1
=4
,
[n(2m + 1)]s
(2m + 1)s
ns
m θ [R
m=0
2
m=0
m=1
(2.9.8)
where N (R, m) denotes the largest integer less than or equal to R 2 /(2m + 1). It
should be noted that the m-sum is cut off at a finite upper limit < [R 2 ].
In the same way one finds that the restricted Madelung constant for the unit
square lattice is
M(R, s) =
(m,n)=(0,0)
N
(R,m)
∞
(−1)m+n
(−1)m
(−1)n
=
4
. (2.9.9)
s
2
2
s
(2m + 1)
ns
(m + n )
m=0
n=1
We therefore have
lim M(R, s) = M(∞, s) = −4(1 − 21−s )ζ (s)β(s).
R→∞
2.9 Commentary: Restricted lattice sums
115
Example 2.2 We now consider a restricted NaCl potential sum.
By following the procedure above we find that
S(R, s) =
=
∞
(−1)l+m+n θ [R 2 − (l + 16 )2 − (m + 16 )2 − (n + 16 )2 ]
l,m,n=−∞
c+∞
1
[(l + 16 )2 + (m + 16 )2 + (n + 16 )2 ]s
dz R 2 z ∞ dt
e
2πi z
t 1−s
0
(s) c−i∞
2
2
2
×
(−1)l+m+n e−(z+t)[(l+1/6) +(m+1/6) +(n+1/6) ] .
l,m,n
However,
∞
l −α(l+1/6)2
(−1) e
=q
1/24
l=−∞
∞
1/3
(−1) (2k + 1)q
k
k(k+1)/2
k=0
where q = e−2α/3 . By noting that S(0, s) = 0 this can be manipulated into the
form
∞
c +i∞
R2
dt
1
du
dy e−ut euy q 1/8
S(R, s) =
(s) 0
t 1−s c −i∞
0
∞
×
(−1)k (2k + 1)q k(k+1)/2 ,
k=0
e−2y/3 . Since
1
the y-integral yields δ[u − 13 k(k + 1) − 36
], the remaining
with q =
integrations are straightforward and produce
S(R, s) = 3s
∞
(−1)k
k=0
2k + 1
[k(k + 1) + 14 ]s
θ [3R 2 −
1
4
− k(k + 1)].
√
Since k√+ 12 + R 3 > 0 the argument of the Heaviside step function is positive
if k < 3R − 12 , so we have the simple formula
√
S(R, s) = 12s
3R−1/2
k=0
(−1)k
.
(2k + 1)2s−1
Now, for Re s > 12 ,
lim S(R, s) = 12s β(2s − 1)
R→∞
√
in agreement with [19]. However, S(R, 12 ) alternates between 2 3 and 0 as R
increases, so that
lim S(R, 12 )
R→∞
does not exist, whereas [19] lims→1/2+0 S(∞, s) =
√
3.
116
Convergence of lattice sums and Madelung’s constant
Example 2.3 We now consider a restricted conical sum. There is a recent
literature on conical theta functions [18] of the form
2
2
q n +m ,
θ K (q) =
(n,m)∈K
where K is a well-structured polyhedral pointed convex cone in the plane. The
Mellin transform Ms [θ K (q)] provides another class of interesting ‘restricted’
lattice sums, of the form
1
.
σ K (2s) =
2
(n + m 2 )s
(n,m)∈K
2.10 Commentary: Other representations for
the Madelung’s constant
No closed form has ever been found for the Madelung constant. This is discussed
at some length in [6], which gives a general discussion of the notion of a closed
form. Many other representations have been proposed, as have various generalizations such as those in [11]. A particularly interesting study is due to Crandall
[10]. Therein, one development starts with a lovely 1986 formula for θ43 (q) due
to Andrews. This is
θ43 (q) = 1 + 4
∞
(−1)n q n
n=1
−2
1 + qn
∞
n=1,| j|<n
(−1) j q n
2− j 2
1 − qn
;
1 + qn
(2.10.1)
see also [4, (9.1.16)].
From (2.10.1) Crandall obtains
M3 (2s) = −4
∞
n,m, p>0
(−1)n+m+ p
− 6α 2 (s).
(nm + mp + pn)s
(2.10.2)
%
&
Here α(s) = 1 − 21−s ζ (s) is again the alternating zeta (or Dirichlet eta) function. Crandall then proceeds to expand the triangular lattice sum in (2.10.2) in
various ways. We comment in passing that there can be only 18 or 19 positive
integers not of the form nm + mp + pn for n, m, p > 0 [5] and the existence of
the nineteenth is ruled out by the Riemann hypothesis. The other eighteen are
1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462 (2.10.3)
and the nineteenth if it existed would exceed 1011 . These are precisely the exceptional discriminants for which no indecomposable binary quadratic form exists
[27]. The non-square elements of (2.10.3) are precisely twice the integers of type
2 listed in [4, p. 293] and 210 leads to Ramanujan’s most famous singular value,
sent in his letter to Hardy and given in [4, (4.6.12)].
2.12 Richard Crandall and the Madelung constant for salt
117
We draw the reader’s attention also to the striking similarity in the structure of
(2.10.2) and (6.3.1), which is a five-dimensional sum with a closed form.
Andrews’ theta function representation of (2.10.1) also led Crandall to a
remarkable integral:
1 2 π
1 + 3 r sin(2θ)−1
1
dθ dr.
M3 = −
%
&%
2
2 &
π 0 0
1 + r sin(2θ)−1 1 + r cos θ 1 + r sin θ
Using another expression for the Madelung constant in [10], Tyagi [28] obtains
the fast converging series
( 18 )( 38 )
1
1 log 2 4π
−
+ √ +
M3 (1) = − −
√
8
4π
3
2 2
π 3/2 2
(−1)m+n+ p (m 2 + n 2 + p 2 )−1/2
,
−2
m,n, p exp 8π m 2 + n 2 + p 2 − 1
(2.10.4)
where the constants before the sum alone give 10 digits of accuracy.
2.11 Commentary: Madelung sums, crystal symmetry,
and Debye shielding
In lattices and plasmas the presence of point-like electrostatic sources or charges
can be accompanied by an exponential screening due to charges of opposite sign
distributed in a cloud around them. This is called Debye screening or shielding
and should be taken into account in physical applications of lattice sums in such
situations. Two recent articles [24, 25] discuss the incorporation of Debye screening effects into the mathematical treatment of lattice sums for structures including
cubic lattices (NaCl), face-centred cubic lattices, the body-centred cubic lattice
(CsCl), and the diamond structure (ZnS). The Dirichlet series which lies behind
much of this,
exp(2πiβr − αr 2 )
,
(2.11.1)
S(β, α) =
3
r ∈Z
r2
is also discussed. From a more mathematical viewpoint, such screening corresponds to the Abel summation of a potentially divergent series [2, 9].
2.12 Commentary: Richard Crandall and the Madelung
constant for salt
A salt crystal is made up of alternating positive and negative electric charges
based on a simple cubic lattice. The electrostatic energy of interaction of this
array of charges was first calculated approximately by Madelung in 1918.
118
Convergence of lattice sums and Madelung’s constant
Essentially it requires the evaluation of the very slowly conditionally convergent
triple sum
α=
(−1)m+n+ p
.
(m 2 + n 2 + p 2 )1/2
(2.12.1)
This was the prototype for the evaluation of arrays of electric charges arranged
on other crystal structures, all referred to as Madelung constants. However, any
reference to the Madelung constant always implies the first example, α. No closed
form for α has ever been found but Richard Crandall initiated novel attempts to
provide better approximations to α with well-known constants of analysis, and
these are reviewed here.
Mathematician–physicist–inventor Richard Crandall
It is with great sadness that the present authors announce the passing of their dear
colleague Richard Crandall, who died on Thursday 20 December 2012, after a
brief bout of acute leukaemia – the week before his 65th birthday on 29 December.
Crandall had a long and colourful career. He was a physicist by training,
studying with Richard Feynman as an undergraduate at the California Institute
of Technology and receiving his Ph.D. in physics at the Massachusetts Institute
of Technology under the tutelage of Victor Weisskopf, the Austrian–American
physicist who discovered what is now known as the Lamb shift and who was
one of the most influential post-war physicists. Richard often commented that he
thought digitally in the fashion of an electrical engineer.
Crandall was for many years at Reed College in Portland, Oregon, where he
directed the Center for Advanced Computation. At the same time, he also worked
for Next Computers (as Chief Scientist), and subsequently for Apple Computers (as Distinguished Scientist) where he was the head of Apple’s Advanced
Computation Group.
Crandall’s research spanned both the theoretical and practical realms: prime
numbers, cryptography, data compression, signal processing, fractals, epidemiology, and, of considerable interest to the present authors, experimental mathematics. He held several patents. He produced many algorithms that are incorporated
into Apple’s products including the iPod and the iPhone. The library of fast
Fourier transforms that was produced by his Advanced Computation Group at
Apple was described by a colleague of ours as ‘miraculously’ fast. He worked on
image processing techniques for Pixar for 13 years, during the last two to remove
artifacts that reportedly could only be seen on Steve Jobs’ personal projector, or
to meet Jobs’ exacting personal requirements that raindrops should look like they
did on celluloid (Richard’s tool was too natural for modern filmgoers).
Indeed, Crandall was a close colleague of Steve Jobs for many years. Crandall
was preparing to write a biography of Jobs, which sadly will not now be written.
2.12 Richard Crandall and the Madelung constant for salt
119
How we did not meet
Unfortunately the author of this commentary section, John Zucker, never had the
pleasure of meeting Richard Crandall in person. We established email connection
over our joint interest in the Madelung constant α. It happened in this way. In
1987 Richard produced two papers on α both of which I was asked to referee.
They were of course both very good and were accepted. However, in one paper I
communicated (as a referee, anonymously) a proof of a conjecture he had made
and, in the other, the result of a sum he had not known. From this he guessed very
quickly who the anonymous reviewer was and we got into contact.
Ever since α first appeared it has been subjected to much analysis, such as (a)
finding ways to evaluate it rapidly or (b) to see whether it might be expressed
in terms of other constants of analysis. It was clear from his paper with Delord
[13] that the problem of evaluating α to many decimal places was easily solved
by Richard, as he gave a value to some 50 decimal places. It is given here to 24
decimal places for future reference:
α = −1.747564594633182190636212 . . .
(2.12.2)
Let me point out immediately that for all practical purposes, such as evaluating
the lattice energy of salt, the first four figures would be enough to compare with
any experimental value. But, just as one only requires π to a mere 35 digits to find
the radius of the universe to the accuracy of the radius of a hydrogen atom, this
does not stop mathematicians calculating it to 10 trillion digits. So, in the paper
with Buhler [12], (b) was tackled.
Richard and the Madelung constant
Richard’s conception was to find an exact expression for α made up of a part
which could be evaluated in terms of well-known constants of analysis plus an
exponentially fast-converging residual sum.
Two of these sums which arise here are
πr 1 (−1)m+n+ p
πr := S+
:=
,
S+ m, n, p;
t
t
r exp(πr /t) + 1
πr 1 (−1)m+n+ p
πr := S−
:=
.
S− m, n, p;
t
t
r exp(πr /t) − 1
Here r := m 2 + n 2 + p 2 , and wherever
appears it will denote summation
over all indices from −∞ to ∞. Note that
πr πr 2πr
+ S−
= −2S−
.
(2.12.3)
S+
t
t
t
Richard found the following exact expression for α:
πr π
α = αC (t) = 4tC(t) −
+ 2S+
,
2t
t
(2.12.4)
120
Convergence of lattice sums and Madelung’s constant
where
(−1)m
t 2
2
π
m + 4t (n 2 + p 2 + q 2 )
(−1)m+n+ p
1 .
=
2π
m 2 + n 2 + p 2 + 4t 2 (q + 12 )2
C(t) : =
Here implies summation over all indices from −∞ to ∞, excluding the case
when all the indices are simultaneously zero.
The residual term 2S+ (πr /t) contributes very little to the value of α and the
smaller t is, the less it contributes. So if C(t) could be evaluated it would provide an approximation to α. Now C(t) involves finding a four-dimensional sum.
Lattice sums are often evaluated using properties of theta functions and because
there are many more theta function identities available in four than in three dimensions, they are often more easily evaluated than three-dimensional sums. This
indeed proved to be the case and the following evaluations (which are analytic
continuations) were provided in [12]:
√
1
2 2 − 2 1
1 3 log 2
1
1
C 4 =
,
C 2 =√ ,
β 8 ,
C(1) = − +
8
4π
π
2
where β(x) is the central beta function defined by
β( p) := B( p, p) =
So, for t = 14 ,
αC ( 14 ) = C( 14 ) − 2π + 2S+3
1
4
2 ( p)
.
(2 p)
√
2 2 − 2 1
β 8 − 2π + 2S+3 14 ,
=
π
(2.12.5)
and we have
√
2 2 − 2 1
β 8 − 2π = −1.747523 . . . ,
π
with the residual series 2S+ (4πr ). Comparing this with (2.12.2) gives agreement
to 4 decimal places, good enough for any practical application. A full account of
Richard’s struggle with α is given in his masterly presentation in [10].
Tyagi’s work
This probably inspired another investigator, Sandeep Tyagi, to a further attack on
α. In this work he naturally came into contact with Richard, who communicated
his work to the author. It encouraged me to find a value for C(t) for t < 14 , namely
√
√
√
3 2 − 6 3 + 6 6 1
1
√
β 8 .
=
C
24
2π
2.12 Richard Crandall and the Madelung constant for salt
121
So, using this in (2.12.4), one has
√
√
√
√
3 2 − 6 3 + 6 6 1
6
β 8 −
= −1.7475621 . . . ,
2π
π
√
with residual 2S+ ( 24πr ). This gives agreement to 5 decimal places with
(2.12.2). However, this is trivial compared with what Sandeep had achieved
in [28]. Sandeep found another exact expression for α, similar in structure to
(2.12.4) but different. It was
πr π
− S−
,
(2.12.6)
αT (t) = T (t) −
6t
t
where
T (t) :=
2t (−1)m+n+ p
.
π
m 2 + n 2 + p 2 + 4t 2 q 2
Sandeep was able to evaluate T (t) for some values of t; thus
T
1
2
1 log 2
,
=− −
2
π
T
1
4
1
1 log 2
+√ .
=− −
4
2π
2
Thus, for t = 1/4,
αT ( 14 ) = − 14 −
2π
log 2
1
+√ −
− 2S− (4πr ).
2π
3
2
(2.12.7)
Very shrewdly, he then took the average of (2.12.5) and (2.12.7) to obtain
√
1
log 2
1
4π
2 2−2
1
αC T = − + √ −
−
+
β
− 2S− (8πr ). (2.12.8)
8 2 2
3
4π
2π
8
It is seen that the residual sum is even smaller.
(This result might have been found directly from a relation discovered later
between C(t) and T (t). This is
4tC(t) + T (t) = 2T (t/2),
hence
2T ( 18 ) = T ( 14 ) + C( 14 ),
which gives (2.12.8) immediately.)
When we evaluate the constants in (2.12.8) we obtain −1.7475645947, in
which the error from α is just 1 in the tenth decimal place. A remarkable
agreement!
Further refinement
As a kind of tribute to Richard, I suggested to Sandeep that we might try to
improve on this. One way of doing this would be to find C(1/8), but we were not
able to do it. However, we could look at evaluating certain terms of 2S− (8πr ).
122
Convergence of lattice sums and Madelung’s constant
Thus Sandeep considered finding this for m = n = 0, n = p = 0, and p = m = 0.
We then get three contributions that are all the same and finish with
2S− (1, 0, 0; 8πr ) = −12
∞
p=1
(−1) p
.
p(e8π p − 1)
(2.12.9)
Now we use the following result in [29]:
∞
p=1
1
4
πc
(−1) p
−
log ,
=−
2π
pc
12
12
kk
p(e
− 1)
(2.12.10)
where k and k are the values of the modulus and complementary modulus of
the complete elliptic integrals K and K of the first kind, which are found when
K /K = c. Here c = 4, and (2.12.9) can be evaluated to give
√
9
45
log( 2−1)−6 log(21/4 +1)−
log 2. (2.12.11)
2
8
When this is added to the constants of (2.12.8) we obtain −1.747564594633175,
which agrees with the α value given by (2.12.2) to 1 in the 14th decimal place.
Thus encouraged, the next set in 2S− (8πr ) was evaluated. There are three terms
of the form, (m = 0, n = p), (n = 0, m = p) and ( p = 0, m = n), leading to
2S− (1, 0, 0; 8πr ) = 4π +
∞
√
1
24 √
.
2S− (1, 1, 0; 8 2πr ) = − √
8
2π
p + 1)
2 p=1 p(e
(2.12.12)
To evaluate this we require a result in [29]:
∞
1
p=1
p(e2π pc + 1)
=−
2K 3 kk πc 1
− log
,
12
6
π3
(2.12.13)
√
with c = 4 2 in (2.12.12). The evaluation of (2.12.12) is somewhat involved but
the result is
√
√
4(1 − b)2 2b(1 + b2 )
2S− (1, 1, 0; 8 2πr ) = 8π + 2 2 log
(1 + b)4
23/4 (1 + b)2 β 18
√
,
(2.12.14)
+ 6 2 log
64π
√
where b := (2 2−2)1/4 . Adding this contribution to (2.12.8) and (2.12.11) gives
−1.7475645946331821917 and the disagreement with α occurs only in the 18th
decimal place.
Finally let us add the contribution of 2S− (8πr ) when m = n = p. In this case,
∞
√
(−1) p
16 √
.
2S− (1, 1, 1; 8 3πr ) = − √
3 p=1 p(e8 3π p − 1)
(2.12.15)
References
123
√
and here c = 4 3; 2S− (1, 1, 1) may be evaluated to give
√
16
(1 − d)4
4√
3 log
, (2.12.16)
2S− (1, 1, 1; 8 3πr ) =
π+
3
9
32(1 + d)2 2d(1 + d 2 )
√
√ 1
where d := (( 3 + 1)/ 8) 2 . When this is added to the constants of (2.12.8),
(2.12.11), and (2.12.14) we obtain −1.74756459463318219063622 where the
agreement with α is now up to 22 decimal places. It is clear how further terms
may be evaluated.
Conclusion
The Madelung constant was only a tiny part of Richard Crandall’s many-faceted
interests. Far more important was his contribution to computational number theory amongst many other accomplishments. The brute force approach used here
of evaluating terms of the residue sums to add more constants of analysis to the
value of α would, I am sure, not have won the approval of Richard. He would have
preferred a more subtle approach on the lines of Sandeep’s derivation of (2.12.8).
Still, I think he would have been amused by the results given here.
References
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3
Angular lattice sums
We discuss here the topic of angular lattice sums, i.e., those which depend on
the angle or angles between the vector linking the origin to lattice points and the
coordinate axes. This topic is an old one, dating back to an 1892 paper by Lord
Rayleigh [26], but is curiously disconnected from the main thread of investigations into lattice sums, as surveyed by Glasser and Zucker [9]. We begin with a
brief account of the history of the sums and go on to give an account of some
of their more recently discovered properties. We use the latter topic to discuss
how properties and formulae for lattice sums may be discovered with the aid of
modern symbolic algebra packages such as Mathematica or Maple. Chief among
the properties of the angular lattice sums in two dimensions that we describe is
their relationship to the Riemann zeta function; selected sums obey the celebrated
Riemann hypothesis.
3.1 Optical properties of coloured glass and lattice sums
The technology of colouring glass by adding to the melt appropriate metals is an
old one, dating back to the time of the ancient Greeks and Romans and predating
the modern field known as plasmonics. Michael Faraday proposed a model of
atoms as being like tiny metal particles and so opened up the question as to what
could be inferred about the properties of atoms from the interaction of light with
various solids. This question was addressed by various investigators, adopting the
approach of supposing that light interacted with the atoms to give them a dipole
moment due to charge separation and that the optical response of the solid could
then be deduced by adding up the periodically arranged response charges.
The most important of the resulting expressions are the Clausius–Mossotti
formulae [4, 20], the Lorentz–Lorenz law [13, 14], and the Maxwell–Garnett
equation [7]. An informative account of the history of this approach may be found
126
Angular lattice sums
in the review article by R. Landauer [12]. Rayleigh commented in his paper that
the treatments of Lorenz and Lorentz, as well as those of previous authors, take
into account only dipole–dipole interactions between particles. His purpose was
to show that this is an approximation, in that for concentrated systems of particles accuracy is only achieved when higher-order effects such as those of induced
quadrupoles and octupoles are taken into account.
In order to do this, Rayleigh considered both electrostatic problems in two
dimensions, involving rectangular and square lattices of cylinders, and also a simple cubic lattice of spheres. The two-dimensional problem is to find a potential
function V (x, y) which satisfies Laplace’s equation inside and outside each cylinder and which obeys continuity of the function and its scaled normal derivative at
cylinder boundaries:
∂ Vint (x, y) ∂ Vext (x, y) σ
=
Vint (x, y)|∂C = Vext (x, y)|∂C ,
,
∂n
∂n
∂C
∂C
(3.1.1)
where ∂/∂n denotes the normal derivative on the boundary of a typical cylinder
and σ denotes a relative transport coefficient such as electrical conductivity or
permittivity (its value inside each cylinder being divided by its value between
cylinders). We write the potential expansion around the central cylinder in the
form of a multipole expansion in polar coordinates (r, θ ):
Vext (r, θ ) =
∞ An r n +
n=1
Bn cos nθ.
rn
(3.1.2)
The choice of a cosine dependence is appropriate if there is an applied external
field along the O x axis and the sum runs only over odd values of n. The coefficients An (and corresponding internal multipole coefficients Cn ) can be related to
the coefficients Bn using the boundary coefficents (3.1.1).
If E 0 denotes the magnitude of the external applied field then the coefficients
Bn are obtained by solving a linear equation called the Rayleigh identity. This
identity states that the part of the expansion (3.1.2) which does not diverge at the
origin, i.e., the part involving the coefficients An , is due to sources on all other
cylinders in the array and the sources at infinity. It may be written as
∞
B2n−1 (2n + 2m − 3)!
S2n+2m−2 B2m−1 = E 0 δn,1 ,
+
(2m − 2)!
a 4n−2
m=1
(3.1.3)
where a denotes the cylinder radius and the lattice sum of order 2n is
(2n − 1)!
1+σ
1−σ
S2n =
R p =(0,0)
cos 2nθ p
.
R 2n
p
(3.1.4)
3.1 Optical properties of coloured glass and lattice sums
127
In this lattice sum, p runs over all the points in the lattice apart from the origin, the
polar coordinates of the pth point being R p and θ p . Given that the lattice sums
can be evaluated, the system of equations (3.1.3) permits the evaluation of the
multipole coefficients B2n−1 . The effective transport coefficient for the array of
cylinders is then given by
2π B1
,
(3.1.5)
σe f f = 1 −
El
where El denotes the local field in the vicinity of the central cylinder.
One subtle question concerning the Rayleigh multipole method is that of the
evaluation of the dipole sum S2 . This sum for a square lattice takes the form
S2 =
p12 − p22
( p1 , p2 )=(0,0)
( p12 + p22 )2
,
(3.1.6)
and thus is conditionally convergent. Rayleigh dealt with this problem by summing over a region much more elongated along the x axis than the y axis. Perrins
et al. [23] showed that the shape-dependent part of the summand in (3.1.7) below
could be absorbed into the expression relating El to E 0 . Another approach is to
introduce a charge-neutralising background distribution, as followed in Chapter 8
and in Poulton et al. [24].
Rayleigh’s approach to the evaluation of the other lattice sums is quite sophisticated and is of sufficient interest to reproduce an outline of it here. He commences
with the sums for the square lattice and notes that they are zero except if the order
is even. The sum of order 2n can then be written for a square lattice of unit period:
1
.
(3.1.7)
S2n =
(m + im)2n
(m,m )=(0,0)
He then notes that sin(ξ − imπ ) is zero when ξ = m π + imπ , m being positive,
negative, or zero. It then follows that, for a suitable constant A,
∞ ξ
ξ
ξ
1−
1−
.
sin(ξ − imπ ) = A 1 −
imπ
imπ + m π
imπ − m π
m =1
(3.1.8)
Putting ξ = 0, we find that A = − sin(imπ ). Rayleigh expands the left-hand side
of (3.1.8), divides by A, and takes the logarithm to obtain
ξ
log [cos ξ − cot(imπ ) sin ξ ] = log 1 −
imπ
∞ ξ
ξ
+ log 1 −
.
+
log 1 −
imπ + m π
imπ − m π
m =1
(3.1.9)
128
Angular lattice sums
He then combines (3.1.9) and its equivalent with m replaced by −m to obtain
sin2 ξ
ξ2
log 1 −
=
log
1
−
(imπ )2
sin2 imπ
∞ ξ2
ξ2
log 1 −
+ log 1 −
.
+
(imπ + m π )2
(imπ − m π )2
m =1
(3.1.10)
This identity can be expanded in a power series in ξ on both sides and the coefficients equated to give expressions for the sums of various orders. The result for
S2 is
∞
1
π2
− 2π 2
S2 =
= π,
(3.1.11)
3
(sinh mπ )2
m=1
where the hyperbolic function sum is obtainable from tables or given by a
symbolic algebra package. Rayleigh’s result for S4 is
∞ π4
1
2
4
+
S4 = 2π
.
(3.1.12)
+
45
sinh4 π m
3 sinh2 π m
m=1
The hyperbolic function series is rapidly convergent: five terms are sufficient to
give the result 3.1512120021538976, accurate to 2 parts in 1014 (the exact value
is ( 14 )8 /(960π 2 ); see Section 4.8).
Given the lattice sums, closed forms may be derived for the effective conductivity or permittivity of a square lattice of circular cylinders of radius a.
The higher-order Rayleigh formula for the square array of circular cylinders
takes into account the multipole coefficients B1 , B3 , B5 , and B7 , and with
T = (1 − σ )/(1 + σ ) is
−1
0.305827 f 4 T
0.013362 f 8
−
2 f, (3.1.13)
σeff = 1 − T + f − 2
T
T − 1.402958 f 8
where f = πa 2 , for a unit lattice spacing, is the area fraction of cylinders in
the array. The numerical values in (3.1.13) take into account the values of the
lattice sums S2 , S4 , S8 , S12 . Their incorporation into the formula for σe f f results in
an expression which takes into account better the multipole interactions between
cylinders in densely packed composites [23].
Rather than continuing with Rayleigh’s argument, we will now follow a more
modern development due to Mityushev [18] and Mityushev and Adler [19].
3.2 Lattice sums and elliptic functions
Mityushev [18] and Mityushev and Adler [19] consider a general two-dimensional
lattice of cylinders, the lattice being generated by fundamental translations in the
3.2 Lattice sums and elliptic functions
129
complex plane ω1 = α > 0 and ω where Im ω = α −1 , so that the parallelogram
cell thus defined has unit area. These authors denote the lattice points m 1 α + m 2 ω
by a linear ordered set e j , with j running from zero to infinity and e0 = 0. With
this notation, the lattice sums of the previous section become
e−2n
(3.2.1)
S2n =
j ,
j
where the primed sum runs over the whole lattice apart from e0 . Let h =
exp(−π/α 2 ). Then Mityushev and Adler [19] give the following expressions for
low-order lattice sums:
∞
π 2 1
mh 2m
−8
,
(3.2.2)
S2 =
α
3
1 − h 2m
m=1
∞
m 3 h 2m
1 π 4 1
+ 16
S4 =
,
(3.2.3)
3 α
15
1 − h 2m
m=1
and
1 π 6
S6 =
15 α
∞
m 5 h 2m
2
− 16
63
1 − h 2m
.
(3.2.4)
m=1
In addition, they give the following recursion formula, which, commencing with
S4 and S6 , gives all higher-order sums:
S2k =
k−2
3
(2m − 1)(2k − 2m − 1)S2m S2(k−m) . (3.2.5)
(2k + 1)(2k − 1)(k − 3)
m=2
Applying the recursion formula (3.2.5) yields lattice sums above S4 and S6 as
combinations of these two:
3 2
5
1
S , S10 =
S4 S6 , S12 =
(18S43 + 25S62 ),
7 4
11
143
3S4 (33S43 + 100S62 )
5S6 (783S43 + 275S62 )
, S18 =
,
=
2431
46189
3S 2 (2178S43 + 12125S62 )
, ...
S20 = 4
508079
S8 =
S16
(3.2.6)
The results (3.2.6) apply to any two-dimensional lattice and give the sums as
functions of the quotient 1/α of the height and the horizontal side length α of the
parallelogram unit cell. They simplify in the case of a square lattice, for which
S6 = S10 = S14 = · · · = 0, so that (3.2.6) becomes
3 2
18 3
99 4
S ,
S ,
S ,
S12 =
S16 =
7 4
143 4
2431 4
6534 4
620730 6
S ,
S ,
=
S24 =
...
508079 4
151915621 4
S8 =
S20
(3.2.7)
130
Angular lattice sums
This list, combined with the value quoted above for S4 , gives highly accurate
values for the square lattice sums up to S24 :
(l, Sl ) = ((4, 3.1512120021538976), (8, 4.2557730353651895),
(12, 3.9388490128279705), (16, 4.0156950330250245),
(20, 3.9960967531762903), (24, 4.000976805303839)).
(3.2.8)
Rayleigh [26] was able to obtain expressions equivalent to those for S4 and S8 and
also highly accurate values for them.
These lattice sums arise in the Taylor series for the Weierstrass ρ(z), ζ (z), and
σ (z) functions [29]. We will now discuss these series and the relationship between
the function log σ (z) and Green’s functions. The Taylor series for log σ (z) is
log σ (z) = log z −
∞
S2n 2n
z .
2n
(3.2.9)
n=2
This function has a logarithmic singularity at z = 0, and also at all the lattice
points e j , since
z
z
z2
1−
exp
σ (z) = z
+ 2
,
(3.2.10)
ej
ej
2e j
j
where the prime over the product symbol denotes the omission of (0, 0). The
Weierstrass invariants for the general lattice are g2 = 60S4 , g3 = 140S6 . The
function σ (z) is related to the Jacobi theta function θ1 (z) by
S2 2
θ1 (z/α)
(3.2.11)
exp
z .
σ (z) = α θ1 (0)
2
We show the behaviour of the real and imaginary parts of this function in
Fig. 3.1. The real part of log σ (z) is a periodic function with a logarithmic singularity at the origin. The imaginary part of log σ (z) has a branch-cut singularity
along the negative real axis in the plots generated in Mathematica and also Maple,
according to their conventions used. The imaginary part of log σ (z) has a linear
gradient along the sides of the unit cell parallel to the y axis, the values there
related by
Im{log[σ (0.5 + i y)]} − Im{log[σ (−0.5 + i y)]} = π(y − sign(y)).
(3.2.12)
The other two Weierstrass functions are given by
∞
1 S2n z 2n−1
ζ (z) = −
z
(3.2.13)
n=2
and
∞
ρ(z) =
1
+
(2n − 1)S2n z 2n−2 .
2
z
n=2
(3.2.14)
3.3 A phase-modulated lattice sum
Re log σ
131
Im log σ
0
3
2
1
0
–1
–2
–3
–1
–2
–3
–4
–5
–0.4
–0.4
–0.2
y
–0.4
–0.4
–0.2
–0.2
0
0
0.2
0.2
0.4
y
x
–0.2
0
0
0.2
0.2
0.4
0.4
x
0.4
Figure 3.1 Plots of the real (left) and imaginary (right) parts of the Weierstrass function
log σ (z), as z ranges over the unit cell of the square lattice.
These Taylor series are derived in an obvious way by differentiating the corresponding series (3.2.9) for log σ (z).
3.3 A phase-modulated lattice sum
We next consider a phase-modulated lattice sum, following a paper by Glasser
[8]. This sum and aspects of its treatment will be of importance in much of the
rest of this chapter. It consists of a sum over the reciprocal lattice corresponding
to a rectangular lattice in two dimensions with periods a and b along the x and y
axes, respectively. The reciprocal lattice is then also rectangular, and consists of
the vectors
u v u, v = 0, ±1, ±2, . . .
(3.3.1)
k = 2π , ,
a b
Then the phase-modulated sum is
=
eik·S
k=0
ks
.
(3.3.2)
Here S is an arbitrary spatial vector lying in the direct lattice; k·S is a dot product.
This lattice sum may also be regarded as a Green’s function, where the operator
is the Laplacian raised to the complex power s/2 (using the ideas of fractional
calculus or distribution theory):
eik·S = i s (ab)
δ(S − R p ) − 1 ,
(3.3.3)
s/2 = i s
k=0
Rp
132
Angular lattice sums
where R p = ( p1 a, p2 b) denotes a general lattice vector in the direct lattice. Here
the Poisson summation formula has been used in going from the first form of
the right-hand side in (3.3.3) to the second. Note that the integral of the righthand side over any unit cell in the direct lattice is zero. In this sense, the Green’s
function is source-neutral, as discussed by Poulton et al. [24].
Glasser [8] starts with the expression
=
a s
2π
(u,v)=(0,0)
e2πi(ux+vy)
[u 2 + (a/b)2 v 2 ]s/2
(3.3.4)
and takes its Mellin transform, to give
(s)
∞
= (a/2π )
dt t s/2−1 [T (t) − 1],
0
T (t) =
exp{−t[u 2 + (a/b)2 v 2 ] + 2πi(ux + vy)},
s
(3.3.5)
u,v
where the sum in T extends over all values of u, v. The next step is to apply the
Poisson summation formula, in this case equivalent to the Jacobi theta function
transformation,
exp(−tu 2 + 2πiux) = (π/t)1/2
u
exp[−(π 2 /t)(u + x)2 ].
(3.3.6)
u
This gives, after separating out the term v = 0 from the double sum,
T (t) − 1 =
e−tu
2 +2πiux
u=0
+ (π/t)1/2
e−(π
2 /t)(u+x)2
e−a
2 v 2 t/b2 +2πivy
.
u;v=0
The double sum here is over all integers u, but over v excluding v = 0.
Consequently,
⎧
a s ⎨ e2πiux
=
2π ⎩
us
u=0
⎫
⎬
π 1/2 2πivy ∞
2
2
2
2
+
e
dt t (s−3)/2 e−π (u−x)/t e−a v t/b . (3.3.7)
⎭
(s)
0
u;v=0
We next use Hobson’s integral, giving the integral in (3.3.7) as a Macdonald
function, in our case of complex order and real argument:
∞
0
dt t s−1 e− pt e−q/t = 2(q/ p)s/2 K s [2( pq)s/2 ],
(3.3.8)
3.3 A phase-modulated lattice sum
assuming p > 0, q > 0, and Re s > 0. This gives the expression for
general form:
=
133
in its final
a s e2πiux
2π 1/2 π b|u + x| (s−1)/2
+
2π
us
(s)
|v|
u=0
u;v=0
2πa
2πivy
. (3.3.9)
|v||u + x| e
×K (s−1)/2
b
The Macdonald functions in (3.3.9) converge exponentially to zero as soon as
the magnitudes of the integers v and u become sufficiently large that the argument exceeds by a factor of around 2 the magnitude of the order, |s − 12 |/2. This
makes this form of expansion suitable for evaluations anywhere in the complex
plane of s.
The expression (3.3.9) simplifies when s is an even integer, as the Macdonald
√
functions of half-integer order reduce to the factor exp(−z)/ z multiplied by a
polynomial. The most important case is, of course, that for s = 2, when can be
regarded as a periodic Green’s function for the Laplace equation. For this case,
K 1/2 (z) =
π
2z
1/2
e−z ,
so that (3.3.9) becomes
⎡
⎤
a 2 e2πiux
1
π
b
⎣
=
e−2π(a/b)|v||u+x|+2πivy ⎦ .
+
2π
a
|v|
u2
u=0
(3.3.10)
(3.3.11)
u;v=0
We take the term u = 0 out of the double sum and then combine positive and
negative values of u and v in the same sums. We introduce the quantities
q = e−π(a/b) ,
α = y + i(a/b)x,
(3.3.12)
and obtain
=
∞
∞
a 2 cos 2π ux
ab 1 2πiαv
(e
+
+ c.c)
4π
v
2π 2
u2
+
u=1
∞ ∞
ab
4π
u=1 v=1
v=1
q 2uv
v
(e2πiαv + e−2πiαv + c.c.),
(3.3.13)
where c.c. denotes the complex conjugate. The v sums can be carried out simply,
since
∞
xv
= − log(1 − x),
v
v=1
(3.3.14)
134
Angular lattice sums
so that
=
∞
a 2 cos 2π ux
ab
|1 − e2πiα |
−
2
2
2π
2π
u
−
u=1
∞
ab
2π
log |(1 − q 2u e2πiα )(1 − q 2u e−2πiα )|.
(3.3.15)
u=1
The first sum is a well-known Fourier series [25], and so (3.3.15) becomes
=
a2 2
ab
(x − x + 16 ) −
log(2e−πax/b | sin π α|)
2
2π
∞
∞
ab 2u
4u −
1 − 2q cos 2π α + q .
log 2π
u=1
(3.3.16)
u=1
The infinite product can be written in terms of the Jacobi theta function θ1 [29]:
∞ 1 − 2q 2u cos 2π α + q 4u = 21/3 cosec z q −1/6
u=1
θ1 (z, q)
.
[θ1 (0, q)]1/3
(3.3.17)
This leads to the final result
θ1 (π α, q) a 2 2 ab
ab
,
(3.3.18)
x −
log 2 −
log 2
6π
2π
[θ1 (0, q)]1/3 n−1/2 q (n+1/2)2 e(2n+1)i z . Here we have corrected
where θ1 (z, q) = ∞
n=−∞ (−1)
a typographical error in the coefficient of the second term on the right-hand side
in [8].
The function (x, y) is shown in Fig. 3.2. It resembles that for the real part of
log σ (z) shown in Fig. 3.1, except in so far as (x, y) is source-neutral whereas
log σ (z) is not.
=
3.4 Double sums involving Bessel functions
We now consider a class of double sums involving Bessel functions, which
in some cases have exact polynomial representations. This class of sums was
investigated [22] for a square lattice of period d:
1 Jl (K h ξ ) i4mψh
e
.
(3.4.1)
Sl,4m,n (ξ ) = n
d
K hn
h=0
Here, l, m = 0, 1, 2, . . . and n = 2, 3, . . . are integers, ξ = d(x, y) with 0 ≤
x, y ≤ 1 and Kh = 2π(h 1 , h 2 )/d is a general reciprocal lattice vector, which has
the polar representation Kh = (K h , ψh ). If ξ is set equal to zero and n = 4m,
these sums reduce to scaled versions of the sums considered by Rayleigh for the
square lattice. We will next exhibit their connection with Glasser’s function .
3.4 Double sums involving Bessel functions
135
Φ
0.5
0.4
0.3
0.2
0.1
0
–0.1
–0.2
–0.4
–0.2
–0.4
–0.2
y
0
0
0.2
0.2
x
0.4
0.4
Figure 3.2 Plot of the Green’s function (x, y) for the Laplace equation (s = 2) as (x, y)
ranges over the unit cell of the square lattice.
To do this, we note that
=
e2πi(ux+vy)/d
ks
k=0
,
(3.4.2)
so that, if we replace k by K h (cos ψh , sin ψh ) and expand the exponential term
using the generating equation for the Bessel functions J p , we find
=
∞
ip
K h =0 p=−∞
J p (ξ K h )ei p(θ−ψh )
,
K hs
(3.4.3)
where (x, y) = ξ(cos θ, sin θ ). For the square lattice, the sums over K h in (3.4.3)
are zero unless p is a multiple of 4, when they are real, so that for s = n we have
=d
n
∞
S4 p,4 p,n (ξ )e4i pθ .
(3.4.4)
p=−∞
Also, the sums S4 p,4 p,n (ξ ) are even under the replacement of p by − p, so that
∞
= d S(0, 0, n) + 2
S4 p,4 p,n (ξ ) cos(4 pθ ) .
n
(3.4.5)
p=1
Thus, the Bessel function sums (3.4.1) over the reciprocal lattice (at least if s is an
integer not smaller than 2) can be seen as the coefficients in the angular Fourier
series in the direct lattice of the Green’s function .
136
Angular lattice sums
3.4.1 Angular-independent sums
We will now indicate the derivation of explicit polynomial solutions for the subset
of Sl,4m,n (ξ ) for which they exist. We start with the case m = 0, for which the
sums which have polynomial expressions have l + n even. (For l + n odd, the
expression for Sl,0,n (ξ ) consists of a term in ξ n−2 and a non-terminating hypergeometric function [22].) The derivation relies on the construction of raising and
lowering operators for the lattice sums. The raising operators rely on the integrals
[28]
a
Jl+1 (ba)
x l+1 Jl (bx) d x = a l+1
(3.4.6)
b
0
and
a
x −l+1 Jl (bx) d x =
0
J−l+1 (ba)
bl−2
− a −l+1
.
l−1
b
2 (l)
From these we find a result which increases both l and n:
ξ
ηl+1 Sl,0,n (η) dη = dξ l+1 Sl+1,0,n+1 (ξ ),
(3.4.7)
(3.4.8)
0
while for n − l > 0 we obtain a result which lowers l and raises n:
ξ
1
d −l+2 η−l+1 Sl,0,n (η) dη = l−1
− dξ −l+1 Sl−1,0,n+1 (ξ ).
2 (l)
(K h d)n−l+2
0
K h =0
(3.4.9)
The series occurring on the right-hand side of equation (3.4.8) is of course familiar
by now:
s s 1
4
L
.
(3.4.10)
=
ζ
−4
(K h d)s
(2π )s
2
2
K h =0
The two lowering operators follow from the following derivative relations for
Bessel functions [28]:
1 l+1
x Jl+1 (bx) = x l+1 Jl (bx),
(3.4.11)
b
and
1 −l+1
x
Jl−1 (bx)
b
= −x −l+1 Jl (bx),
(3.4.12)
where the prime denotes differentiation with respect to x. Hence, the operation
which lowers both l and n is
(3.4.13)
Sl−1,0,n−1 (ξ ) = dξ −l ξ l Sl,0,n (ξ ) ,
and that which lowers n and increases l is
Sl+1,0,n−1 (ξ ) = −dξ l ξ −l Sl,0,n (ξ ) .
(3.4.14)
3.4 Double sums involving Bessel functions
137
The final recurrence relation of use is that derived from the recurrence relation
for Bessel functions:
2l
Jl (z) = Jl−1 (z) + Jl+1 (z),
z
which gives
2d
Sl+2,0,n (ξ ) = (l + 1)
ξ
(3.4.15)
Sl+1,0,n+1 (ξ ) − Sl,0,n (ξ ).
(3.4.16)
To start the use of the recurrence relations, we use the results (3.3.18) and
(3.4.5), which enable us to evaluate the lowest-order sum, S(0, 0, 2). Comparing
these two, we find that
θ1 (π α, q) 1
ξ2
1
log 2 −
log ,
(3.4.17)
S0,0,2 (ξ ) = 2 −
6π
2π
4d
[θ1 (0, q)]1/3 a.i.p.
where the notation ‘a.i.p.’ means that we are to take the part of the expression that
is independent of the angle θ . Note that
log |θ1 (π α, q)|a.i.p. = log
ξ
+ log 2π θ1 (0, q),
2d
leading to
1
ξ
S0,0,2 (ξ ) = −
log
+ ω0 +
2π
2d
ξ
2d
(3.4.18)
2
,
(3.4.19)
where
ω0 = −
1
[3 log π + 4 log 2 + 2 log θ1 (0, e−π )] −0.318895593319287.
6π
(3.4.20)
The (ξ/2d)2 term in (3.4.19) is equivalent to the ‘jellium’ used by [11] and others,
and fulfils the function of removing the net charge from the Green’s function.
Substituting (3.4.19) into (3.4.5) (with n = 2) and using the relation (3.4.29), we
obtain another expression for [24], free of the shape-dependent sum S2 which
has caused difficulties in previous formulations.
Commencing with the result (3.4.19), we can use the raising and lowering operators from (3.4.8), (3.4.9), (3.4.13), and (3.4.14) to determine explicit expressions
for any sum Sl,0,n (ξ ) with l + n even. Examples of the results of this process are
ξ
ξ
ξ
1 ξ 3
1
1
log
+ ω0 +
+
S1,0,3 = −
, (3.4.21)
2π 2d
2d
4π
2d
2 2d
S2,0,4 (ξ ) = −
1
4π
ξ
2d
2
log
2
1
ξ
ξ
3
1 ξ 4
+
ω0 +
+
,
2d
2
8π
2d
6 2d
(3.4.22)
138
and
1
S0,0,4 (ξ ) =
2π
Angular lattice sums
ξ
2d
2
2
λ
ξ
ξ
1
1 ξ 4
+
log
− ω0 +
−
,
2d
2π
2d
4 2d
24π 2
(3.4.23)
where in the last of these we have used ζ (2) = π 2 /6 and L −4 (2) = λ 0.915965594177219, λ being the Catalan constant.
The last two results that we give are for m = 0:
1 ξ 2
1
−
(3.4.24)
S2,0,2 (ξ ) =
4π
2 2d
and
1
S3,0,3 (ξ ) =
8π
ξ
2d
1
−
6
ξ
2d
3
.
(3.4.25)
3.4.2 Angular-dependent sums
For m = 0, the recurrence relations (3.4.8), (3.4.13), (3.4.14), and (3.4.16) remain
unaltered. We have to reconsider the relation (3.4.9), which becomes
ξ
eimψh
d −l+2 η−l+1 Sl,m,n (η) dη = l−1
− ξ −l+1 d Sl−1,m,n+1 (ξ ),
2 (l)
(K h d)n−l+2
0
K h =0
(3.4.26)
provided that n − l ≥ 0. The double sum in (3.4.26) was shown in Movchan et al.
[21] to take the following value for the square lattice:
e4imψh
e4imψh
1
1
2π
(4m)
σs
δs,2 ,
=
=
−
(K h d)s
(2π )s
(2π )s
4m
(h 21 + h 22 )s/2
K =0
(h ,h )=(0,0)
h
1
2
(3.4.27)
where the Kronecker delta contribution in the last term arises since the following
limit has been taken:
eimψh e2πiR·(h 1 ,h 2 )/d
σs(m) = lim
.
(3.4.28)
R→0
(h 21 + h 22 )s
(h ,h )=(0,0)
1
2
This exceptional contribution is related to the sum S2 discussed in the first section
of this chapter. Thus, the required recurrence relation for angular sums is
ξ
d −l+2
2π
(4m)
−l+1
σn−l+2 −
δl,n
η
Sl,4m,n (η) dη = l−1
4m
2 (l)(2π )n−l+2
0
− ξ −l+1 dSl−1,4m,n+1 (ξ ),
(3.4.29)
for n − l ≥ 0. It will be observed then that all recurrence relations preserve the
angular order 4m of the sums.
3.4 Double sums involving Bessel functions
139
The simplest form for an angular sum occurs when l = 4m and n = 2, and is
(4m)
S4m,4m,2
σ
= 4m
16π m
4m
ξ
.
d
(3.4.30)
(4m)
Here the σ4m
are just the S4m discussed in the first two sections of this chapter.
Another useful starting point for the application of the recurrence formulae is
S2,4m,2 (ξ ) =
2m−1
m (2m + 1)s (−2m + 1)s (4m) ξ 2s+2 1 ξ 2
σ2s+2
−
.
2π
s!(s + 2)!
d
8 d
s=0
(3.4.31)
Here, (x)n = (x + n)/ (x) denotes the Pochhammer symbol. For example, if
m = 1 then we have
1 (4) π ξ 2 4 (4) ξ 4
σ2 −
S2,4,2 (ξ ) =
− σ4
,
(3.4.32)
π
2
2d
π
2d
2
(4)
(4)
(4)
where σ2 4.0784511611614. Let us define σ2 = σ2 − π/2. Then, using
(3.4.32) as our starting point, the recurrence relations give the following explicit
forms for m = 4 sums:
3
ξ
ξ
12
1 2
2 (4) ξ 5
(4)
(4)
− σ2
+ σ4
,
(3.4.33)
σ
S1,4,3 =
2d
π
2d
π
2d
(2π )2 2
3
5
ξ
ξ
1 2
1
S3,4,3 =
− σ4(4)
,
(3.4.34)
σ2(4)
3π
2d
π
2d
2
4
ξ
ξ
1 2
1 2
1 (4) ξ 6
(4)
(4)
σ
S2,4,4 =
−
+
,
(3.4.35)
σ
σ
2d
3π 2
2d
2π 4
2d
8π 2 2
2
ξ
1 2
22
6 (4) ξ 4
(4)
(4)
σ
S0,4,2 =
−
+
,
(3.4.36)
σ
σ
π 2
2d
π 4
2d
4π 2 2
2
4
ξ
ξ
1
1 2
1 2
2 (4) ξ 6
(4)
(4)
(4)
σ
S0,4,4 =
σ −
+
−
,
σ
σ
2d
2π 2
2d
3π 4
2d
16π 4 4
4π 2 2
(3.4.37)
3
5
ξ
ξ
ξ
1 2
1
1 2
(4)
(4)
(4)
−
S1,4,5 =
σ4
+
σ2
σ2
4
2
2d
2d
6π
2d
16π
8π
7
1 (4) ξ
σ
−
.
(3.4.38)
6π 4
2d
By the time n has reached 5, the sums (3.4.1) over the reciprocal lattice are easily
evaluated by direct summation, enabling the easy verification of formulae of the
type just presented.
140
Angular lattice sums
3.5 Distributive lattice sums
We next consider lattice sums which are connected with the theory of distributions
or generalized functions, following the discussion in the paper by McPhedran
et al. [17]. Such distributive lattice sums are associated with sets of local sources
at each lattice point, whose amplitudes are linked by a phase factor which varies
in plane-wave fashion from one lattice point to the next. This phase factor arises
very often in solid state physics, where it is called the Bloch factor. The lattice
sums that we discuss are associated with solutions of the Helmholtz equation
with wavenumber k, and we denote the Bloch vector which determines the phase
factor by k0 . If R p denotes the pth lattice vector, in polar form (R p , φ p ) in two
dimensions, the basic lattice sums are given by
(1)
Hl (k R p )eilφ p eik0 ·R p .
(3.5.1)
Sl (k, k0 ) =
p=0
(1)
Here the term Hl (kr )eilφ is a solution of the Helmholtz equation which behaves
like an outgoing cylindrical wave for large r , and whose source is located at r = 0.
(1)
The Hankel function of the first kind Hl (kr ) is the sum of a Bessel function Jl
(1)
and a Neumann function Yl , so that Hl (kr ) = Jl (kr ) + iYl (kr ), and in terms of
the last functions we write Sl = SlJ + i SlY .
We first evaluate the lattice sums SlJ (k, k0 ). The Poisson summation formula
in two dimensions is
2 2π
e−is·R p =
δ(s − Kh ),
(3.5.2)
d
p
h
where we are dealing with a square lattice of period d and the reciprocal lattice vectors Kh have Cartesian components (2π h 1 /d, 2π h 2 /d), h 1 and h 2 being
integers. We put s = k − k0 , so that
1+
e
−i(k−k0 )·R p
=
p=0
2π
d
2 δ(k − k0 − Kh ).
(3.5.3)
h
With θ = arg k, we have from the Bessel function expansion
e
−ik·R p
=
∞
(−i)l Jl (k R p )eil(φ p −θ) ,
(3.5.4)
l=−∞
which combined with (3.5.3) gives
1+
∞
l=−∞
l −ilθ
(−i) e
p=0
Jl (k R p )e
ilφ p ik0 ·R p
e
=
2π
d
2 δ(k − k0 − Kh ).
h
(3.5.5)
3.5 Distributive lattice sums
141
We define phased reciprocal lattice vectors Qh by Qh = k0 + Kh and put Q h =
|k0 +Kh |, θh = arg Qh . The right-hand side of (3.5.5) can then be rewritten to give
2 ∞
δ(k − Q h )δ(θ − θh )
2π
.
(−i)l e−ilθ
Jl (k R p )eilφ p eik0 ·R p =
1+
d
k
p=0
l=−∞
h
(3.5.6)
Multiplying equation (3.5.6) by exp(inθ ) and integrating over θ from 0 to 2π
yields
2πi l SlJ (k, k0 ) = −δl,0 +
δ(k − Q h )eilθh .
(3.5.7)
kd 2
h
SlJ
The
are thus distributive in nature, their distributive nature being concentrated
around the lines Q h = k in reciprocal space. These are called light lines in the
literature on photonic band gap materials and correspond to the dispersion relation
of plane waves.
We next consider the lattice sums SlY . We obtain an expression for these
by comparing two different expressions for a quasi-periodic Green’s function
satisfying
δ(r − R p )eik0 ·R p .
(3.5.8)
(∇ 2 + k 2 )G(r) = −2π d 2
p
These are a spatial form,
G(r) =
iπ (1)
H (k|r − R p )|eik0 ·R p ,
2 p 0
(3.5.9)
and a spectral form,
G(r) = −
2π eiQh ·r
.
d2
Q 2h − k 2
h
(3.5.10)
We expand the Hankel function terms in (3.5.8) using Graf’s addition theorem
[1], after separating out the term for p = 0. We expand the exponential term in
the spectral form using (3.5.4). The two terms are compared by integrating over
the angular direction of the arbitrary vector r from 0 to 2π , to give
(1
Sl Jl (kr ) = −H0 (kr )δl,0 −
or
(1
Sl Jl (kr ) = −H0 (kr )δl,0 −
4i l+1 Jl (Q h r )eilθh
d2
Q 2h − k 2
h
(3.5.11)
1
4i l+1 Jl (Q h r )eilθh
1
.
+
2Q h
Qh + k
Qh − k
d2
h
(3.5.12)
142
Angular lattice sums
The representation (3.5.12) has one sum which is non-singular on the right-hand
side but a second sum which has singularities at values h such that Q h = k.
These singularities are treated using the Plemelj formula, which expresses them
as a combination of a delta function and a Cauchy principal value:
1
1
= iπ δ(x) + P
.
(3.5.13)
x − i0
x
We then obtain
Sl Jl (kr ) = −J0 (kr )δl,0 − iY0 (kr )δl,0
−
1
4i l+1 Jl (Q h r )eilθh
1
.
)
+
P
+
iπ
δ(k
−
Q
h
2Q h
Qh + k
Qh − k
d2
h
(3.5.14)
Comparing equations (3.5.14) and (3.5.6), we find
SlY Jl (kr )
1
1
4i l+1 Jl (Q h r )eilθh
+P
.
= −Y0 (kr )δl,0 − 2
2Q h
Qh + k
Qh − k
d
h
(3.5.15)
This equation shows that the SlY are singular at light lines, with first-order poles
there which are to be treated numerically using the Cauchy principal value.
The representation (3.5.15) is of formal use in deriving properties of the lattice
sums SlY . An alternative form may be derived using the free parameter r and
integrating over it to improve the convergence of the series for large values of Q h
[3]. The result after m integrations is
SlY Jl+m (kr )
m
1 (m − n)! 2 m−2n+2
= − Ym (kr ) +
δl,0
π
(n − 1)! kr
− 4i l
h
n=1
k
Qh
m
Jl+m (Q h r )eilθh
.
Q 2h − k 2
(3.5.16)
3.6 Application of the basic distributive lattice sum
The expression for the distributive lattice sums (3.5.7) may be used to generate
expressions for double sums related by the Poisson summation formula. We can
multiply this expression by any real or complex function of the wavenumber k:
p=0
f (k)Jl (k R p )eilφ p eik0 ·R p = − f (k)δl,0 +
2πi l f (Q h )
δ(k − Q h )eilθh .
Qh
d2
h
(3.6.1)
3.6 Application of the basic distributive lattice sum
143
We then eliminate the delta functions on the right-hand side of (3.6.1) by
integration over k:
∞
f (k)Jl (k R p )dk eilφ p eik0 ·R p
p=0
0
∞
=−
0
f (k) dk δl,0 +
2πi l f (Q h ) ilθh
e .
Qh
d2
(3.6.2)
h
To apply the general formula (3.6.2), we need f (k) to be such that the two integrals over k are evaluable in closed form, either as ordinary integrals or in the
sense of generalized functions. Three other applications of the formula are given
in [17] but we will focus here on only one, which involves complex powers of k.
We rely on results obtained using the methods of the theory of distributions
from Chapter 7 of the book by D. S. Jones [10] and, in particular,
∞
∞
( 12 β)!2β+2 π
|x|β e−iα·x dx = 2π
r β+1 J0 (αr ) dr =
, (3.6.3)
(− 12 β − 1)!α β+2
−∞
0
valid when β = 2m and β = −2 − 2m (m = 0, 1, 2, . . .). Hence, if s = 2m − 1
or −1 − 2m,
∞
12 + 12 s 2s
r s J0 (αr ) dr = .
(3.6.4)
0
12 − 12 s α 1+s
By differentiation under the integral sign in (3.6.4) and the use of Bessel function
recurrence relations, we obtain Weber’s integral,
∞
12 + 12 l + 12 s 2s
r s Jl (αr ) dr = ,
(3.6.5)
0
12 + 12 l − 12 s α 1+s
provided that 1 + l ± s is not a negative even integer.
If we put f (s) = k 2s−1 in (3.6.2) and use Weber’s integral, the result is
straightforward for l = 0:
⎛
⎞
l
ilθ0
eik0 ·R p eilφ p
eilθh
e
2πi
⎠.
= 2 (1 − s + 12 l) ⎝ 2−2s +
22s−1 (s + 12 l)
R 2s
d
k0
Q 2−2s
p
h
p=0
h=0
(3.6.6)
The result is not so straightforward for l = 0 because of the term
2s ∞
∞
k
k 2s−1 dk =
,
2s 0
0
(3.6.7)
which is undefined at either the lower or the upper limit. This difficulty is dealt
with in the original article [17] by the introduction of raising and lowering operators, partial derivative combinations with respect to the components of the vector
144
Angular lattice sums
k0 . When these operate, they either reduce or increase the integer l by one, while
decreasing 2s to 2s −1. Through the use of these operators, it can be demonstrated
that the nugatory term (3.6.7) is to be replaced by zero. Using this, we obtain
⎛
⎞
eik0 ·R p
1
2π
1
⎠.
22s−1 (s)
= 2 (1 − s) ⎝ 2−2s +
(3.6.8)
R 2s
d
k0
Q 2−2s
p
h
p=0
h=0
One valuable application of the equations (3.6.6) and (3.6.8) is to deduce functional equations for lattice sums. If we let the length of the vector k0 tend to zero,
then the sums over the pairs (R p , φ p ) for the square array may be identified with
those over the pairs (Q h , θh ) (we use here the fact that the square lattice is identical, apart from a rescaling, to its reciprocal lattice). When |k0 | → 0, for the
square lattice the angular lattice sums in (3.6.6) tend to zero unless l is divisible
by 4. If we define the following dimensionless sum,
(2l + s) e4ilφ p
(2l + s)
=
π 2l+s
(R p /d)2s
π 2l+s
( p1 + i p2 )4l
,
2 + p 2 )s+2l
(
p
1
2
p=0
( p1 , p2 )=(0,0)
(3.6.9)
we find that it satisfies for all integers l the functional equation
G 4l (s) =
G 4l (s) = G 4l (1 − s).
(3.6.10)
For a fixed value of the integer l, G 4l (s) is an analytic function of the complex
variable s, and the functional equation shows that G 4l (s) is real on the critical
line Re s = 12 . This fact will be important in the discussion of the location of its
zeros which follows. We also note the correspondence between the G 4n (s) and
the sums S4n mentioned above in connection with the work of Lord Rayleigh:
S4n = G 4l (2n).
3.7 Cardinal points of angular lattice sums
We continue the discussion of angular lattice sums using elements of the article
[15]. There, the following notation is introduced for a dimensionless form of the
lattice sums occurring in equation (3.6.6):
m
(k0 ) = d 2s
σ2s
exp(imφ p )
exp(ik0 · R p )
R 2s
p
(3.7.1)
p=0
and
μl2s (k0 ) =
exp(ilθh )
exp(ilθ0 )
l
=
+ ρ2s
(k0 ).
2s
(d Q h )
(k0 d)2s
d Qh
(3.7.2)
3.7 Cardinal points of angular lattice sums
145
l (k ) is continuous as k → 0, while μl (k ) is certainly well defined as
Here ρ2s
0
0
2s 0
k0 → 0 if Re s < 0. These sums are connected by (3.6.8):
l
(k0 ) = π 2s−1 i l
σ2s
( 12 l + 1 − s)
( 12 l + s)
μl2−2s (k0 ).
(3.7.3)
We now replace k0 by the dimensionless quantity κ = (k0 d)/(2π ) and
investigate the limit for l = 0 as s → 1 by putting s = 1 + 12 δ, where δ → 0:
π 1+δ (− 12 δ)
σ20 (k0 ) = lim
δ→0
(1 + 12 δ)
0
(κ, θ0 )].
[eδ ln κ + ρ−δ
(3.7.4)
0 (k ) which may be derived by expanding the factor
We use an expression for ρ2s
0
2s
1/Q h in a Taylor series for small κ:
ρ2−2s (κ, θ0 ) =
(0)
σ2−2s
+
∞ (l + 1 − s) 2
l!(1 − s)
l=1
(0)
κ 2l σ2l+2−2s
∞
∞
cos 4nθ0 (4n + q + 1 − s)(q + 1 − s) (4n)
σ2q+4n+2−2s κ 2q+4n .
+2
q!(q + 4n)!
[(1 − s)]2
n=1
q=0
(3.7.5)
Here we have introduced the notation
(m)
m
σ2s
= lim σ2s
(k0 ),
(3.7.6)
k0 →0
and we recall that for the square lattice m must be divisible by 4 for a non-zero
(4n)
value to occur, while σ4n = S4n as defined in Section 3.2.
We need to keep in (3.7.4) only terms which remain non-zero as δ → 0. The
prefactor is of the form 2π/δ, and so we need to keep in the single sum over l in
(3.7.5) only the term for l = 1. In the double sum, we need to retain only q = 0.
Hence
∞
σ20 (κ, θ0 )
2
cos 4nθ0
−2π
(0) δ
(0)
S4n κ 4n .
= lim
+ κ 2 σ2−δ
−δ
1 + δ ln κ + σ−δ
δ→0 δ
4
4n
n=1
(3.7.7)
(0)
Since δ2s
has a simple pole at s = 1, we have
(0)
σ2−δ −2π
+D
δ
(3.7.8)
&
δ%
C + ln 2π ,
2
(3.7.9)
and
(0)
σ−δ = −1 +
146
Angular lattice sums
giving
σ20 (κ, θ0 ) = 2π ln κ − π(C + ln 2π ) + π 2 κ 2 +
∞
π cos 4nθ0
S4n κ 4n . (3.7.10)
2
n
n=1
In these expressions,
C = ln
( 14 )4
8π 2
0.783188785414,
π
(3.7.11)
D = π 2γ + ln − C 2.5849817596.
2
In this section and in Chapter 4 we will be interested in the evaluation of the
lattice sums σ20 (κ, θ0 ) at points in the Brillouin zone (the central unit cell in the
reciprocal lattice), where (κx , κ y ) = (κ cos θ0 , κ sin θ0 ) takes rational values, of
the form (m x /n, m y /n). The particular question we will consider now is the location of the special or cardinal points where the σ20 (κ, θ0 ) take simple closed forms
consisting of a single term similar to that encountered in the Lorenz–Hardy result:
(0)
σ2s = 4ζ (s)L −4 (s).
(3.7.12)
Using the formula (3.6.7), the closed forms for the σ20 (κ, θ0 ) at the cardinal points
also give simple expressions for the displaced lattice sums
1
.
(3.7.13)
Sm x ,m y ;n (2s) =
2
[(h 1 + m x /n) + (h 2 + m y /n)2 ]s
h 1 ,h 2
We can locate the cardinal points by exploiting their analytic form, which must
(0)
consist of a product of σ2s with an algebraic prefactor [15]. It is easy to derive
0 (κ , κ ) following the derivation used in
Macdonald function series for the σ2s
x
y
0 (κ , κ ) over
Section 3.3 [15]. These can be used to construct contour plots of σ2s
x
y
the Brillouin zone. We note that if s is chosen to correspond to a zero of either
0 (κ , κ ) must have a zero at every cardinal point. This
ζ (s) or L −4 (s) then σ2s
x
y
procedure is used in Fig. 3.3, where s is chosen to coincide with the first zero of
L −4 (s) (at s 12 + 6.020948905i).
The cardinal points evident in Fig. 3.3 were isolated by methods other than
numerical strategies apart from the innermost points, which occur at (κx , κ y ) =
1
3
3
1
, ± 10
) and (κx , κ y ) = (± 10
, ± 10
). The forms of the lattice sums at the
(± 10
cardinal points of the square lattice are:
S0,1;2 (2s) = 22s+1 (1 − 2−s )L 1 (s)L −4 (s),
0
σ0,1;2
(2s) = 23−2s (1 − 2s−1 )L 1 (s)L −4 (s),
(3.7.14)
S1,1;2 (2s) = 2s+2 (1 − 2−s )L 1 (s)L −4 (s),
0
σ1,1;2
(2s) = 23−s (1 − 2s−1 )L 1 (s)L −4 (s),
(3.7.15)
S1,1;4 (2s) = 4 (2 − 1)L 1 (s)L −4 (s),
s
s
0
σ1,1;4
(2s) = −22−3s (2s − 2)L 1 (s)L −4 (s).
(3.7.16)
3.8 Zeros of angular lattice sums
147
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0 (κ , κ ) for s equal to the first zero of L (s), as
Figure 3.3 Plot of the lattice sum σ2s
x y
−4
(κx , κ y ) ranges over the first quadrant of the Brillouin zone of the square lattice. The dots
represent cardinal points.
These expressions may be derived easily from the basic form (3.7.12). The cardinal point related to n = 5 was reported by Zucker [30], the lattice sum being
evaluated by him using theta function techniques:
S1,2;5 (2s) = (5s − 1)L 1 (s)L −4 (s),
0
σ1,2;5
(2s) = (−1 + 51−s )L 1 (s)L −4 (s).
(3.7.17)
Note the pattern in these sums: the prefactor in S is non-zero at s = 1, while the
first-order zero of the prefactor in σ 0 cancels the first-order pole of ζ (s) there. For
n = 10 we have
S1,3;10 (2s) = (5s−1 − 1)(2s−1 − 1)L 1 (s)L −4 (s),
0
σ1,3;10
(2s) = (1 − 21−s )(1 − 51−s )L 1 (s)L −4 (s).
(3.7.18)
Note that, for s → 1, both lattice sums tend to zero since the prefactor has a
1
3
, ± 10
) and (κx , κ y ) =
double zero there. In fact, the points (κx , κ y ) = (± 10
3
1
0
(± 10 , ± 10 ) all lie on the contour σ2 (κx , κ y ) = 0.
The data of Fig. 3.3 pertains to the region where κ > 0.1. The function
0 (κ , κ ) = 0 is highly oscillatory near κ = 0, being dominated by the first
σ2s
x
y
term κ 2s−2 . However, this first term has modulus 1/κ for Re s = 12 , and so there
are no cardinal points in the region where κ is non-zero and small.
3.8 Zeros of angular lattice sums
The celebrated Riemann hypothesis affirms that all the non-trivial zeros of ζ (s) lie
on the critical line Re s = 1/2. It has received extensive numerical confirmation
148
Angular lattice sums
but has yet to be proved analytically. The generalized Riemann hypothesis affirms
that, for every Dirichlet L-function associated with a Dirichlet character, once
again the non-trivial zeros lie on the critical line. Thus, assuming the generalized
(0)
Riemann hypothesis to hold, the basic sum σ2s will have all its non-trivial zeros
on Re s = 12 , as will the sums σ20 (κx , κ y ) at all cardinal points.
It is an open question whether comparable statements can be made for other
points in the Brillouin zone or for other sums. However, some very significant results have been obtained, which should motivate further investigations.
The results pertain to the following sums, which already have been introduced
in (3.6.9):
C(1, 4m; s) =
cos 4mθ p , p
1 2
p1 , p2
( p12 + p22 )s
= C(2, 2m; s) − S(2, 2m; s),
(3.8.1)
where
C(n, m; s) =
cosn mθ p , p
1 2
( p12 + p22 )s
S(n, m; s) =
sinn mθ p , p
1 2
.
( p12 + p22 )s
(3.8.2)
These sums are considered in [16], where it is shown that all elements of the
families of sums C(2, 2m; s) and S(2, 2m; s) can be represented as finite combinations of sums from the family C(1, 4m; s). Numerical evidence indicates
that the first two families of sums have zeros off the critical line, while the third
has its non-trivial zeros on the critical line. Note that, from (3.8.1), C(0, 1; s) =
4L 1 (s)L −4 (s).
These remarks are supplemented by two theorems, proofs of which may be
found in McPhedran et al. [16].
p1 , p2
,
p1 , p2
Theorem 3.1 Suppose that C(0, 1; s) obeys the Riemann hypothesis. Then
C(1, 4m; s) obeys the Riemann hypothesis for any positive integer m. Conversely,
if C(1, 4m; s) obeys the Riemann hypothesis for some m then C(0, 1; s)) obeys
the Riemann hypothesis.
Theorem 3.2 If Theorem 3.1 holds then the distribution functions for the zeros
NC1,4 ( 12 , t) and NC0,1 ( 12 , t) must agree in all terms which go to infinity with t.
Thus the validity of the Riemann hypothesis for ζ (s) and L −4 (s) together is equivalent to the validity of the Riemann hypothesis for all members of the class of
angular sum C(1, 4m; s). Furthermore, the number of zeros of ζ (s) and L −4 (s)
together up to any real number t agrees with that for any angular sum C(1, 4m; s)
in so far as all terms which diverge with t as it goes to infinity are concerned.
3.9 Commentary: Computational issues of angular lattice sums
149
3.9 Commentary: Computational issues
of angular lattice sums
(1) The angular lattice sums defined in (3.8.2) can be evaluated numerically from
the subset of sums
p12n
( p1 , p2 )
( p12 + p22 )s+n
C(2n, 1; s) =
.
(3.9.1)
Using the Mellin transform and Hobson’s integral, as described at the end of
Section 1.2, we find
√
2 π (s + n − 12 )ζ (2s − 1)
C(2n, 1; s) =
(s + n)
∞ ∞ s
p2 s−1/2
8π
+
(π p1 p2 )n K s+n−1/2 (2π p1 p2 );
(s + n)
p1
p1 =1 p2 =1
(3.9.2)
here the term involving ζ (2s − 1) comes from the axial contribution (where
Hobson’s integral does not converge), in this case when p2 = 0.
This sum is exponentially convergent since for large arguments the modified
Bessel function of the second kind is approximated by
(
π −z
e .
K n+1/2 (z) ∼
2z
In practice, the sum requires p1 and p2 to be large enough that 2π p1 p2 exceeds
|s − 12 | by a factor 2 or so before rapid convergence sets in. All the other angular
sums defined in (3.8.2) can be obtained by using trigonometric addition formulae
to obtain appropriate combinations of the C(2n, 1; s) (including of course the
sums for n = 0). Note also that we have the limit
lim C(2n, 1; s) = 2ζ (s).
n→∞
(2) The same technique involving the Mellin transform, Poisson summation, and
Hobson’s integral can be used to obtain good approximations for a large class of
sums. For instance, we consider
S(F, G; s) =
p0 , p1
F( p1 )
,
[G( p1 ) + p02 ]s
where p0 runs over the integers and p1 could stand for a multi-index; the prime
excludes those values of p1 , p0 for which the denominator of the summand is
zero. We require G to be a non-negative function, increasing as | p1 | → ∞ at least
as rapidly as | p1 |a for some a > 0; moreover, we require that p1 F( p1 )/G( p1 )s
150
Angular lattice sums
converges absolutely for some range of s. Then, we have
1
2π 2s− 2 ( 12 − s)ζ (1 − 2s)
S(F, G; s) =
F( p1 )
(s)
p1 : G( p1 )=0
√
1
π (s − 2 )
F( p1 )
+
(s)
G( p1 )s−1/2
p1 : G( p1 )=0
+
s−1/2
p0
F( p1 ) √
(s)
G( p1 )
p0 =1 p1 : G( p1 )=0
%
&
× K s−1/2 2π p0 G( p1 ) .
4π s
∞
(3.9.3)
(3) A useful numerical technique for investigating factorizations of lattice sums
is a consideration of their expansions in the direct summation region Re s 1.
In this region, the terms of a sum may be readily evaluated in order of powers of
increasing distance from the origin. Possible expansions involving say products
of Dirichlet L functions may then be tested against the terms arising in the sum in
question, using the product rule for the products:
∞
∞
∞
ep
cn dm
=
,
s
s
n
m
ps
n=1
m=1
(3.9.4)
p=1
where on the right-hand side e p is the sum of cn dm over all positive integer pairs
(n, m) such that nm = p. The possibility of prefactors of algebraic form as well
as products of Dirichlet L-functions has to be taken into account. The knowledge
of the form of the functional equation satisfied by the target sum is valuable in
assembling appropriate factorization elements. This technique will be useful in
obtaining some factorizations in Chapter 4.
3.10 Commentary: Angular lattice sums and the
Riemann hypothesis
(1) An interesting feature of the lattice sums C(1, 4m; s) is the distribution of
gaps between their zeros on the critical line, for m = 0. The distribution functions of the gaps between zeros for C(1, 4; s) and C(0, 1; s) are quite different,
particularly for small gaps. The former fits well the Wigner surmise [2, 6] for the
Gaussian unitary ensemble:
−4S 2
9S 2
,
(3.10.1)
P(S) = 2 exp
π
π
where P(S) denotes the probability of finding a gap S in a distribution of gaps
whose mean has been scaled to unity. This distribution gives a probability that
goes to zero as S 2 for small gaps. By contrast, for C(0, 1; s) [2], the product
3.10 Commentary: Angular lattice sums and the Riemann hypothesis 151
factorization gives an uncorrelated superposition of two unitary ensemble sets,
with no clear diminution in the probability as S tends to zero. Thus, it seems
possible that the distribution of gaps between zeros might be a good indicator of
whether simple factorizations of lattice sums are possible or not.
(2) It is of interest to consider non-dimensionalized sums over rectangular lattices
in two dimensions of the form [17]
σ (0) (d1 , d2 ; s) = (d1 d2 )s
=
( p12 d12
( p1 , p2 )=(0,0)
( p12 d1 /d2
p=0)
1
+ p22 d22 )s
1
.
+ p22 d2 /d1 )s
(3.10.2)
This sum may be put into a form suitable for numerical evaluation by the use of
Mellin transform techniques, in the usual way. We introduce a general form for
sums over Macdonald functions:
K (n, m; s; λ) =
s−1/2
∞
p2
( p1 p2 π )n K m+s−1/2 (2π p1 p2 λ). (3.10.3)
p1
p1 , p2 =1
The quantity λ is introduced, as usual, as an adjunct in analysis – for example,
differentiation with respect to it could generate sums over the derivatives of Macdonald functions and help one to deduce recurrence relations among the sums. It
is also useful to introduce two combinations of zeta functions:
(s)ζ (2s) (s − 12 )ζ (2s − 1)
+
,
4π s
4π s−1/2
(s)ζ (2s) (s − 12 )ζ (2s − 1)
T− (s) =
−
.
4π s
4π s−1/2
T+ (s) =
(3.10.4)
The first of these is symmetric under s → 1 − s and the second is antisymmetric.
It may be shown that both satisfy the Riemann hypothesis; this was proved for
T− (s) by Taylor [27]. We then have
σ
(0)
s−1
√ (s − 12 )
d2
d1 s
ζ (2s − 1)
(d1 , d2 ; s) = 2ζ (2s)
+2 π
d2
(s)
d1
)
s
d1
8π
d1
.
(3.10.5)
+
K 0, 0; s;
(s) d2
d2
A similar expansion may be derived by, for example, differentiating with
respect to d1 /d2 in the rightmost expression in (3.10.2), while regarding d2 /d1
152
Angular lattice sums
as constant, or following the Mellin transform route:
p12
p=0)
( p12 d1 /d2 + p22 d2 /d1 )s+1
s
(s + 12 ) √
d2
π ζ (2s − 1)
(s + 1)
d1
)
d1
8π s
d1
. (3.10.6)
+
K 1, 1; s;
(s + 1) d2
d2
=2
(3) The preceding treatment allows us to obtain explicit analytic expressions for
Macdonald function sums. We put d1 = d2 = 1 and then have sums which we
have already encountered on the left-hand sides of (3.10.5) (see also (3.10.2)) and
(3.10.6). From these we get
K (0, 0; s; 1) =
(s)ζ (s)L −4 (s) (s)ζ (2s) (s − 12 )ζ (2s − 1)
−
−
(3.10.7)
2π s
4π s
4π s−1/2
and
K (1, 1; s; 1) =
1
s(s)ζ (s)L −4 (s)
1 (s − 2 )ζ (2s − 1)
−
(s
−
)
.
2
4π s
4π s−1/2
(3.10.8)
K (0, 0; s; 1) is real on the critical line and has zeros both on this line and off it
[17] whereas K (1, 1; s; 1) is complex on the critical line.
(4) We can find linear combinations of K (0, 0; s; 1) and K (1, 1; s; 1) which are
real on the critical line:
K (1, 1; s; 1) − 12 s K (0, 0; s; 1) = 14 T+ (s) + 12 (s − 12 )T− (s)
(3.10.9)
and
K (1, 1; s; 1) −
1
2 (s
−
1
2 )K (0, 0; s; 1)
(0)
(s)σ2s
1
1
s
−
T− (s).
=
+
2
2
32π s
(3.10.10)
Both these functions have their zeros on the critical line and close to those of
T− (s), since the term (s − 12 ) multiplying this function means that for large t it
dominates its partner function. It is easy to furnish a first-order estimate for the
shift of the zero away from the corresponding zero of T− (s).
(5) Horizontal Riemann curves
Fig. 3.4 plots the modulus of the function Z
defined by (x, y) → |ζ (x + i y)| for 15 < x < 45 and 1 < y < 100.
It is instructive to plot Z (x, y) as a function of x on the critical strip [0, 1] for
various values of y; see Fig. 3.5. (Compare what happens when the roles of x and
y are reversed.) As it turns out, the monotonicity exhibited in the graph of level
curves of Z (·, y) implies the Riemann hypothesis. This conclusion is given in [31]
3.10 Commentary: Angular lattice sums and the Riemann hypothesis 153
Figure 3.4 The modulus of the zeta function.
3
2.5
2
1.5
1
0.5
0
10
15
20
25
30
35
40
t
Figure 3.5 The first few zeros of ζ on the critical line.
and was discovered while the first author of that paper was teaching an undergraduate complex variable class. Much related material and many illustrations are to
be found in Conrey’s survey [5].
(6) The Riemann hypothesis as a random walk Recall the Liouville lambda
function λ(n) := (−1)r (n) , where r (n) counts the number of prime factors with
repetition in n. Fig. 3.6 shows λ plotted as a two-dimensional random walk,
154
Angular lattice sums
0
–200
–400
–600
–800
–1000
–1200
–1400
–1600
–1800
0
200
400
600
800
1000
1200
Figure 3.6 The Riemann hypothesis as a random walk on Liouville’s function.
found by walking with steps (±1, ±1) through pairs of points of the sequence
{λ(2k), λ(2k + 1)}nk=1 . The fact that n steps of the walk stay in a box with side
√
of length roughly n implies the Riemann hypothesis: that all non-real zeros of
the ζ -function have a real part equal to one-half (giving the critical line). The first
few zeros are plotted in Fig. 3.6. Note that
∞
λ(n)
n=1
ns
=
ζ (2s)
.
ζ (s)
References
[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with
Formulas, Graphs and Mathematical Tables. Dover, New York, 1972.
[2] E. Bogomolny and P. Leboeuf. Statistical properties of the zeros of zeta functions –
beyond the Riemann case. Nonlinearity, 7:1155–1167, 1994.
[3] S. K. Chin, N. A. Nicorovici, and R. C. McPhedran. Green’s function and lattice
sums for electromagnetic scattering by a square array of cylinders. Phys. Rev. E,
49, 1994.
[4] R. Clausius. Die mechanische Behandlung der Electricität. Braunschweig, Vieweg,
1879.
[5] J. B. Conrey. The Riemann hypothesis. Not. Amer. Math. Soc., 50:341–353, 2003.
[6] B. Dietz and K. Zyczkowski. Level-spacing distributions beyond the Wigner surmise.
Z. Phys. Condens. Matter, 84:157–158, 1991.
References
155
[7] J. C. M. Garnett. Colours in metal glasses and in metallic films. Phil. Trans. Roy.
Soc. London, 203:385–420, 1904.
[8] M. L. Glasser. The evaluation of lattice sums. III. Phase modulated sums. J. Math.
Phys., 15:188–189, 1974.
[9] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and
Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980.
[10] D. S. Jones. Generalized Functions. McGraw-Hill, New York, 1966.
[11] C. Kittel. Introduction to Solid State Physics. Wiley, New York, 1953.
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[13] H. A. Lorentz. The Theory of Electrons. B. G. Teubner, Leipzig, 1909. Reprint:
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[14] L. Lorenz. Wiedemannsche Ann., 11:70, 1880.
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investigation of two-dimensional static array sums. J. Math. Phys., 48:033501, 2007.
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hypothesis and the zero distribution of angular lattice sums. Proc. Roy. Soc. London
A, 467:2462–2478, 2011.
[17] R. C. McPhedran, G. H. Smith, N. A. Nicorovici, and L. C. Botten. Distributive and
analytic properties of lattice sums. J. Math. Phys., 45:2560–2578, 2004.
[18] V. V. Mityushev. Transport properties of doubly-periodic arrays of circular cylinders.
Z. Angew. Math. Mech., 77:115–120, 1997.
[19] V. V. Mityushev and P. M. Adler. Longitudinal permeability of spatially periodic
rectangular arrays of circular cylinders. I. A single cylinder in the unit cell. Z. Angew.
Math. Mech., 82:335–345, 2002.
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Scienze Residente in Modena, 24:49–74, 1850.
[21] A. B. Movchan, N. A. Nicorovici, and R. C. McPhedran. Green’s tensors and lattice
sums for elastostatics and elastodynamics. Proc. Roy. Soc. London A, 453:643–662,
1997.
[22] N. A. Nicorovici, C. G. Poulton, and R. C. McPhedran. Analytical results for a class
of sums involving Bessel functions and square arrays. J. Math. Phys., 37:2043–2052,
1996.
[23] W. T. Perrins, D. R. McKenzie, and R. C. McPhedran. Transport properties of regular
arrays of cylinders. Proc. Roy. Soc. London A, 369:207–225, 1979.
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Green’s functions for periodic problems and their equivalents in electromagnetism.
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Cambridge University Press, Cambridge, 1946.
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4
Use of Dirichlet series with
complex characters
4.1 Introduction
In Section 1.4, Dirichlet series with real characters were introduced in order to find
2
2 −s
closed forms for lattice sums of the type ∞
m,n=0,0 (am + bmn + cn ) . Here,
Dirichlet series with complex characters and their application to finding closed
forms for other types of lattice sums will be discussed. We first recapitulate the
properties of real L-series given in Chapter 1 with additional definitions of certain
properties and some new notation. This will enable a straightforward comparison
to be made with the complex L-series to be considered immediately after that.
4.2 Properties of L-series with real characters
The properties of L-series with real characters are well known and have been
given elsewhere; see e.g., Ayoub [2], Zucker and Robertson [17]. Elementary
L-series (modulo k) are given by
L k (s) =
∞
χ k (n)
n=1
ns
;
(4.2.1)
k will be referred to as the period or order of the L-series. The quantity χ k is
called a character; it is a multiplicative function and periodic in k and is defined
as follows. For integers m, n,
χ k (1) = 1,
χ k (n + k) = χ k (n),
χ k (n) = 0
χ k (mn) = χ k (m)χ k (n),
if gcd(k, n) = 1.
(4.2.2)
It was shown by Dickson [6] that χ k can only assume values which are the φ(k)th
roots of unity, where φ(k) is the Euler totient function giving the number of positive integers less than k which are relatively prime to k. Thus characters may be
real or complex.
158
Use of Dirichlet series with complex characters
For real L-series χ k (n) can only be ±1. All real L-series then divide into two
types, according to whether χ k (k − 1) = ±1. If χ k (k − 1) = +1 then the series
will be said to have positive parity. For such series the signs of the characters will
be mirrored exactly in the midpoint of the series. If χ k (k −1) = −1 then the signs
of the characters will be mirrored by opposite signs. Theorems given in Ayoub
[2] show that the only possible characters are given by the Legendre–Jacobi–
Kronecker symbol: χ k (n) = (n|k). The number of independent real-character
L-series is then found to be as follows. Let P = 1 or let P be a product of all
different primes, i.e., odd and square-free; then
(1)
(2)
(3)
(4)
if k = P there is just one primitive L-series,
if k = 4P there is just one primitive L-series,
if k = 8P there are two primitive L-series,
if k = 2P, 2α P, where α > 3 or P is not square free, there is no primitive
L-series.
The definition of a primitive L-series is complicated. A good account is given by
Ayoub [2], and it is discussed fully in Zucker and Robertson [17].
The parity of a real L-series is determined as follows.
(1)
(2)
(3)
(4)
(5)
If k
If k
If k
If k
If k
= P ≡ 1 (mod 4) the L-series has positive parity.
= P ≡ 3 (mod 4) the L-series has negative parity.
= 4P with P ≡ 3 (mod 4) the L-series has positive parity.
= 4P with P ≡ 1 (mod 4) the L-series has negative parity.
= 8P there is an L-series of each parity.
The suffix k will be signed according to whether χ k (k − 1) = ±1. If s is
a positive integer then explicit formulae for L −k (2s − 1) and L k (2s) may be
established. They are
k
(1 − n/k)
(−1)s−1 22s−2 π 2s−1 B
χ −k (n) 2s−1
L −k (2s − 1) =
√
(2s
− 1)!
k
n=1
(4.2.3)
and
L k (2s) =
k
(−1)s−1 22s−1 π 2s B (1 − n/k)
χ k (n) 2s
,
√
(2s)!
k
n=1
(4.2.4)
where Bs (x) are the Bernoulli polynomials. As both n and k are positive integers
and Bs (1 − n/k) are rational numbers then, for s a positive integer,
√
L −k (2s − 1) = R(k) k π 2s−1 ,
√
L k (2s) = R (k) k π 2s ,
4.2 Properties of L-series with real characters
159
where R(k) and R (k) are rational numbers. It is also known that
2h(k)
L k (1) = √ log 0 ,
k
where h(k) is the class number of the binary quadratic√form of discriminant k
and 0 is the fundamental unit of the number field Q( k). Some examples of
real-character L-series are given below. It is pertinent to introduce here another
notation for L-series, which emphasizes their periodicity. Let
(k, l : s) := (k, l) =
∞
n=0
1
;
(kn + l)s
(4.2.5)
then
L k (s) =
k−1
χ k (l)(k, l).
(4.2.6)
l=1
Note that (k, l : s) may be expressed as
∞
1 1
1 l
=
ζ s, ,
ks
(n + l/k)s
ks
k
(4.2.7)
n=0
where ζ (s, q) denotes the Hurwitz zeta function. (For numerical and symbolic
computations of (k, l), see the commentary in Chapter 8.)
The Riemann zeta function is thus expressed as follows:
1
1
1
(4.2.8)
ζ (s) = L 1 (s) = 1 + s + s + s + · · · = (1, 1).
2
3
4
For the Catalan (or Dirichlet) β function we have
1
1
1
β(s) = L −4 (s) = 1 − s + s − s + · · · = (4, 1) − (4, 3).
3
5
7
(4.2.9)
Other real-character L-series which will be used here are
L −3 (s) = (3, 1) − (3, 2),
L 5 (s) = (5, 1) − (5, 2) − (5, 3) + (5, 4),
L −7 (s) = (7, 1) + (7, 2) − (7, 3) + (7, 4) − (7, 5) − (7, 6),
L −8 (s) = (8, 1) + (8, 3) − (8, 5) − (8, 7),
L 8 (s) = (8, 1) − (8, 3) − (8, 5) + (8, 7),
L 12 (s) = (12, 1) − (12, 5) − (12, 7) + (12, 11),
L −20 (s) = (20, 1) + (20, 3) + (20, 7) + (20, 9)
− (20, 11) − (20, 13) − (20, 17) − (20, 19),
L 28 (s) = (28, 1) + (28, 3) − (28, 5) + (28, 9) − (28, 11) − (28, 13)
− (28, 15) − (28, 17) + (28, 19) − (28, 23) + (28, 25) + (28, 27).
160
Use of Dirichlet series with complex characters
In addition to these there is always one further real L-series of period k formed
by all the characters χ k (n) equal to 1; this can be shown to be equal to (1 −
pn−s )L 1 (s), where the pn are all the different prime factors of k. This is illustrated
below for k = 5 by making use of the expansion property of (k, l); thus
(k, l) = (2k, l) + (2k, k + l) = (3k, l) + (3k, k + l) + (3k, 2k + l) = · · · .
So
(1, 1) = (5, 1) + (5, 2) + (5, 3) + (5, 4) + (5, 5),
but (5, 5) = 5−s (1, 1) and so
(5, 1) + (5, 2) + (5, 3) + (5, 4) = (1, 1) − (5, 5) = (1 − 5−s )L 1 .
As explained in Chapter 1, these real L-series are used to express double sums
given by
Q(a, b, c; s) = Q(a, b, c) =
∞
(am 2 + bmn + cn 2 )−s
(4.2.10)
m,n=0,0
as sums of products of pairs of real L-series of opposite parity. (Please note the
change of notation from S used in Chapter 1 to Q as used here. The symbol S will
be required for another type of sum later on). Some examples are
Q(1, 0, 5) = L 1 L −20 + L −4 L 5 ,
Q(1, 1, 2) = 2L 1 L −7 ,
2Q(5, 2, 10) = 1 + 71−2s L 1 L −4 − L −7 L 28 .
In particular, many Q(1, 0, λ) have been solved by expressing them as the
Mellin transforms of the product of the Jacobian θ3 (q) functions with different
arguments. Thus
∞
∞
1
s−1
Q(1, 0, λ) =
t
exp −m 2 t − λn 2 t dt
(s) 0
m,n=0,0
∞
1
=
t s−1 θ3 (q)θ3 (q λ ) − 1 dt, where e−t = q
(s) 0
and
θ3 (q) =
∞
2
q n = 1 + 2q 2 + 2q 4 + 2q 9 + · · · .
n=−∞
By expressing θ3 (q)θ3 (q λ ) − 1 as a Lambert series where possible, the integral
is easily evaluated in terms of L-series. A long list of such results may be found
in Table 1.6 and supplemented by others in Zucker and Robertson [18]. Not all Q
can be thus factored, and the possible use of complex L-series has long been contemplated by Zucker as possibly providing a vehicle for exposing non-susceptible
Q to possible factorization. However, it turns out that complex L-series are useful
4.3 Properties of L-series with complex characters
161
in factorizing another kind of two-dimensional lattice sum, so an account of these
will now be given.
4.3 Properties of L-series with complex characters
In order to compare real L-series with complex L-series, we first list all the possible L-series with both real and complex characters for periods k = 1−10 and
for k = 16. For k a prime, these are found by taking each φ(k)th root of unity in
turn and using the rules for characters to produce a given series, always starting
with +1 for the first term. If k has factors then these factors also have period k
and will reduce the number of independent L-series of period k. This will become
evident from the following.
For
For
k = 1,
k = 2,
φ(1) = 1;
φ(2) = 1;
(1, 1) = L 1 .
(2, 1) = (1 − 2−s )L 1 .
For k = 3, φ(3) = 2 as 1 and 2 are relatively prime to 3. There are two possible
order-3 L-series:
(3, 1) + (3, 2) = (1 − 3−s )L 1 ,
(3, 1) − (3, 2) = L −3 .
For k = 4, φ(4) = 2 as 1 and 3 are relatively prime to 4. There are two possible
order-4 L-series:
(4, 1) + (4, 3) = (1 − 2−s )L 1 ,
(4, 1) − (4, 3) = L −4 .
For k = 5, φ(5) = 4 and the four roots of unity are ±1 and ±i, giving four
order-5 L-series:
(5, 1) + (5, 2) + (5, 3) + (5, 4) = (1 − 5−s )L 1 ,
(5, 1) − (5, 2) − (5, 3) + (5, 4) = L 5 ,
(5, 1) + i(5, 2) − i(5, 3) − (5, 4) = L i−5 ,
(5, 1) − i(5, 2) + i(5, 3) − (5, 4) = L −i
−5 .
For k = 6, φ(6) = 2, as 1 and 5 are relatively prime to 6. There are just two
possible L-series, both real and of lower order than 6:
(6, 1) + (6, 5) = (1 − 2−s )(1 − 3−s )L 1 ,
(6, 1) − (6, 5) = (1 + 2−s )L −3 .
For k = 7, φ(7) = 6 as 1, 2, 3, 4, 5, 6 are relatively prime to 7. The sixth
roots of unity are 1, −1, ω, −ω, ω2 , −ω2 where ω = exp(iπ/3). The six possible
L-series are
162
Use of Dirichlet series with complex characters
(7, 1) + (7, 2) + (7, 3) + (7, 4) + (7, 5) + (7, 6) = (1 − 7−s )L 1 ,
(7, 1) + (7, 2) − (7, 3) + (7, 4) − (7, 5) − (7, 6) = L −7 ,
(7, 1) + ω2 (7, 2) + ω(7, 3) − ω(7, 4) − ω2 (7, 5) − (7, 6) = L ω−7 ,
2
(7, 1) − ω(7, 2) − ω2 (7, 3) + ω2 (7, 4) + ω(7, 5) − (7, 6) = L −ω
−7 ,
(7, 1) + ω2 (7, 2) − ω(7, 3) − ω(7, 4) + ω2 (7, 5) + (7, 6) = L ω7 ,
2
(7, 1) − ω(7, 2) + ω2 (7, 3) + ω2 (7, 4) − ω(7, 5) + (7, 6) = L −ω
7 .
For k = 8, φ(8) = 4 as 1, 3, 5, 7 are relatively prime to 8. The four order-8
L-series are all real:
(8, 1) + (8, 3) + (8, 5) + (8, 7) = (1 − 2−s )L 1 ,
(8, 1) − (8, 3) + (8, 5) − (8, 7) = L −4 ,
(8, 1) + (8, 3) − (8, 5) − (8, 7) = L −8 ,
(8, 1) − (8, 3) − (8, 5) + (8, 7) = L 8 .
For k = 9, φ(9) = 6 as 1, 2, 4, 5, 7, 8 are relatively prime to 9. There are six
order-9 L-series:
(9, 1) + (9, 2) + (9, 4) + (9, 5) + (9, 7) + (9, 8) = (1 − 3−s )L 1 ,
(9, 1) − (9, 2) + (9, 4) − (9, 5) + (9, 7) − (9, 8) = L −3 ,
(9, 1) − w (9, 2) − w(9, 4) + w(9, 5)] + w2 (9, 7) − (9, 8) = L −ω
−9 ,
2
2
(9, 1) + w(9, 2) + w2 (9, 4) − w 2 (9, 5) − w(9, 7) − (9, 8) = L ω−9 ,
(9, 1) + w2 (9, 2) − w(9, 4) − w(9, 5) + w2 (9, 7) + (9, 8) = L ω9 ,
2
(9, 1) − w(9, 2) + w2 (9, 4) + w 2 (9, 5) − w(9, 7) + (9, 8) = L −ω
9 .
For k = 10, φ(10) = 4 as 1, 3, 7, 9 are relatively prime to 10. The four possible
L-series are all of lower order than 10:
(10, 1) + (10, 3) + (10, 7) + (10, 9) = (1 − 2−s )(1 − 5−s )L 1 ,
(10, 1) − (10, 3) − (10, 7) + (10, 9) = (1 + 2−s )L 5 ,
(10, 1) − i(10, 3) + i(10, 7) − (10, 9) = (1 − i2−s )L i−5 ,
(10, 1) + i(10, 3) − i(10, 7) − (10, 9) = (1 + i2−s )L −i
−5 .
For k = 16, φ(16) = 8 as 1, 3, 5, 7, 9, 11, 13, 15 are relatively prime to 16.
So, there are eight L-series. As φ(k) gets larger the depiction of L-series in (k, l)
symbols can become unwieldy. It is thus convenient here to introduce a slightly
modified notation, namely a signed (k, l) symbol defined by
(k, l)+ := (k, l) + (k, k − l),
(k, l)− := (k, l) − (k, k − l).
(4.3.1)
4.3 Properties of L-series with complex characters
163
This enables us to halve the number of symbols required to show an L-series. It
will be seen that positive-parity series are described entirely by (k, l)+ symbols
and negative-parity series by (k, l)− terms. Thus for k = 16 we have
(16, 1)+ + (16, 3)+ + (16, 5)+ + (16, 7)+ = (1 − 2−s )L 1 ,
(16, 1)− − (16, 3)− + (16, 5)− − (16, 7)− = L −4 ,
(16, 1)− + (16, 3)− − (16, 5)− − (16, 7)− = L −8 ,
(16, 1)+ − (16, 3)+ − (16, 5)+ + (16, 7)+ = L 8 ,
(16, 1)− + i(16, 3)− + i(16, 5)− + (16, 7)− = L i−16 ,
(16, 1)− − i(16, 3)− − i(16, 5)− + (16, 7)− = L −i
−16 ,
(16, 1)+ + i(16, 3)+ − i(16, 5)+ − (16, 7)+ = L i16 ,
(16, 1)+ − i(16, 3)+ + i(16, 5)+ − (16, 7)+ = L −i
16 .
These results illustrate most properties that have been observed for complex
L-series and allow us to make the following conjectures regarding them.
(1) The number of positive-parity series always equals the number of negativeparity series.
(2) The number of complex-character positive-parity L-series is even and they
divide into pairs of complex conjugates. The same is true for complexcharacter L-series with negative parity.
(3) The first and the last terms of all L-series are always real.
(4) Knowing the parity of the series and the first non-real term then, using the
properties of characters given in (4.2.2), the coefficients of all the other
terms may be established. This allows us to create a concise notation for
complex L-series. The subscript gives the parity and period whilst the superscript gives the coefficient of the first non-real term, and this specifies the
given series completely.
(5) The number of (k, l) symbols needed to specify a given L-series equals the
number of L-series for a given k. Thus every (k, l) is expressible as a linear
combination of L-series of period k.
We note from the above that for k = 6, 10 the inclusion of complex L-series
does not yield any L-series with these periods, and this has also been found for
k = 14 and 18. So, part of the statement made in Section 4.2 regarding real
L-series seems to hold for complex L-series, namely, if k is equal to twice an
odd number then no L-series of such a period exists. It would appear that this is
a result of the fact that φ(2n) = φ(n) if n is an odd number. However, whereas,
for real L-series, if k is a perfect square > 4 then no L-series of such period are
found, it is seen here that, for such k, complex L-series of that period do exist.
164
Use of Dirichlet series with complex characters
It is necessary to point out here that all the statements about real L-series made
in the previous section have been proved whereas the conjectures made in this
section about complex L-series have not been proved.
4.4 Expressions for displaced lattice sums in closed
form for j = 2−10
Displaced lattice sums were introduced by McPhedran et al. [11, 12]. They are of
the form
r 2
p 2
+ n+
S( p, r, j; s) := S( p, r, j) =
m+
j
j
m,n
−s
(4.4.1)
and are related to the Green’s function connected with sums over the square
lattice. There are also associated phased lattice sums,
exp[2πi(mp/j + nr/j)]
,
(m 2 + n 2 )s
σ ( p, r, j; s) = σ ( p, r, j) =
(4.4.2)
m,n=0,0
the displaced and phased lattice sums being connected by the Poisson summation
formula.
It is easily seen that S( p, r, j) = S( j − p, j −r, j), so all the independent S for
a given j will be found by allowing both p and r to take on all values up to and
including j/2. Then the displaced lattice sums may be expressed as the following
Mellin transform:
S( p, r, j) =
m,n
=
j 2s
(s)
m + p/j
0
∞
2
1
+ n + r/j
m,n
j 2s
( jm + p)2 + ( jn + r )2
, exp − ( jm + p)2 t + ( jn + r )2 t dt.
∞
t s−1
2 s =
m,n=−∞
(4.4.3)
The exponential sums are disjoint and each can be evaluated separately. Then,
letting e−t = q and writing
∞
q ( jm+ p) = θ ( j, p),
2
m=−∞
we have
S( p, r, j) =
j 2s
(s)
0
∞
t s−1 θ ( j, p)θ ( j, r ) dt.
(4.4.4)
s
4.4 Expressions for displaced lattice sums in closed form for j = 2−10 165
For small j, θ ( j, p) may be expressed in terms of θ3 -functions, see [15], and
we list the results for j = 2, 3, 4, 6:
% 2&
θ (2, 1) = θ3 (q) − θ3 q 4 ,
θ ( j, 0) = θ3 q j ,
1
1
θ3 (q) − θ3 q 9 ,
θ3 (q) − θ3 (q 4 ) ,
θ (4, 1) =
θ (3, 1) =
2
2
4
16
,
θ (6, 0) = θ3 (q 36 ),
θ (4, 2) = θ3 (q ) − θ3 q
1
θ3 (q) − θ3 (q 4 ) − θ3 (q 9 ) + θ3 (q 36 ) ,
θ (6, 1) =
2
1
4
θ3 (q ) − θ3 (q 36 ) ,
θ (6, 2) =
θ (6, 3) = θ3 (q 9 ) − θ3 (q 36 ).
(4.4.5)
2
For these j values all the independent S( p, r, j) may be expressed in terms of
Q(1, 0, λ), using
∞
∞
1
t s−1
exp −(m 2 + λn 2 )t dt,
Q(1, 0, λ) =
(s) 0
m,n=0,0
∞
1
=
t s−1 θ3 (q)θ3 (q λ ) − 1 dt, where e−t = q. (4.4.6)
(s) 0
Thus we obtain
S(0, 1, 2) = 22s Q(1, 0, 4) − 4−s Q(1, 0, 1) ,
S(1, 1, 2) = 22s Q(1, 0, 1) − 2Q(1, 0, 4) + 4−s Q(1, 0, 1) ,
S(0, 1, 3) =
S(1, 1, 3) =
S(0, 1, 4) =
S(0, 2, 4) =
S(1, 1, 4) =
S(1, 2, 4) =
32s
2
32s
2
42s
2
42s
2
42s
2
42s
2
Q(1, 0, 9) − 9−s Q(1, 0, 1) ,
Q(1, 0, 1) − 2Q(1, 0, 9) + 9−s Q(1, 0, 1) ,
Q(1, 0, 16) − 4−s Q(1, 0, 4) ,
2−2s Q(1, 0, 4) − 2−4s Q(1, 0, 1) ,
Q(1, 0, 1) − 2Q(1, 0, 4) + 4−s Q(1, 0, 4) ,
Q(1, 0, 4) − 4−s Q(1, 0, 1) − Q(1, 0, 16) − 4−s Q(1, 0, 4) ,
(4.4.7)
S(0, 0, 6) = Q(1, 0, 1),
1 −2s
6 Q(1, 0, 1) − 3−2s Q(1, 0, 4) − 2−2s Q(1, 0, 9)
S(0, 1, 6) =
2
+ Q(1, 0, 36)] ,
166
Use of Dirichlet series with complex characters
S(1, 1, 6) =
62s
(1 + 2−2s + 3−2s + 6−2s )Q(1, 0, 1)
4
− 2(1 + 3−2s )Q(1, 0, 4) − 2(1 + 2−2s )Q(1, 0, 9)
+ 2Q(4, 0, 9) + 2Q(1, 0, 36) ,
32s −2s
−3 Q(1, 0, 1) + Q(1, 0, 9) ,
S(0, 2, 6) =
2
62s
−2−2s (1 + 3−2s )Q(1, 0, 1) + (1 + 3−2s )Q(1, 0, 4)
S(1, 2, 6) =
4
+ 21−2s Q(1, 0, 9) − Q(4, 0, 9) − Q(1, 0, 36) ,
32s (1 + 3−2s )Q(1, 0, 1) − 2Q(1, 0, 9) ,
S(2, 2, 6) =
2
S(0, 3, 6) = −Q(1, 0, 1) + 2−2s Q(1, 0, 4),
S(1, 3, 6) =
S2, 3, 6) =
62s
(−3−2s − 6−2s )Q(1, 0, 1) + 2 × 3−2s Q(1, 0, 4)
2
+ (1 + 2−2s )Q(1, 0, 9) − Q(4, 0, 9) − Q(1, 0, 36) ,
62s −2s
6 Q(1, 0, 1) − 3−2s Q(1, 0, 4) − 2−2s Q(1, 0, 9)
2
+ Q(4, 0, 9) ,
S(3, 3, 6) = 22s (1 + 2−2s )Q(1, 0, 1) − 2Q(1, 0, 4) .
(4.4.8)
Of the six different Q(a, b, c) which appear in the preceding displaced sums, the
values of four have been found in terms of Dirichlet series with real characters.
They are
Q(1, 0, 1) = 4L 1 (s)L −4 (s),
Q(1, 0, 4) = 2(1 − 2−s + 21−2s )L 1 (s)L −4 (s),
Q(1, 0, 9) = (1 + 31−2s )L 1 (s)L −4 (s) + L −3 (s)L 12 (s),
Q(1, 0, 16) = (1 − 2−s + 21−2s − 21−3s + 22−4s )L 1 (s)L −4 (s) + L −8 (s)L 8 (s).
The functions Q(1, 0, 36) and Q(4, 0, 9) cannot be individually found (for
Q(1, 0, 36) see also the commentary in Section 4.6). However, it will be seen
that ± [Q(1, 0, 36) + Q(4, 0, 9)] appear together in three of the cases considered
and it may be shown, via the theory of which numbers can be represented by the
binary quadratic forms (m 2 + 36n 2 ) and (4m 2 + 9n 2 ), that1
Q(1, 0, 36) + Q(4, 0, 9) = (1 − 2−s + 21−2s )(1 + 31−2s )L 1 (s)L −4 (s)
+ (1 + 2−s + 21−2s )L 3 (s)L 12 (s).
So, we are able to express eight of the ten possible S( p, r, 6) in terms of known
Dirichlet series. All the results for j = 2 − 4 and j = 6 are listed below.
1 We are grateful to Mark Watkins, University of Bristol, for obtaining this result for us.
4.4 Expressions for displaced lattice sums in closed form for j = 2−10 167
For j = 2,
S(0, 1, 2) = 22s+1 (1 − 2−s )L 1 (s)L −4 (s),
S(1, 1, 2) = 2s+2 (1 − 2−s )L 1 (s)L −4 (s).
For j = 3,
32s
2
32s
S(1, 1, 3) =
2
S(0, 1, 3) =
(1 − 3−2s )L 1 (s)L −4 (s) + L −3 (s)L 12 (s) ,
(1 − 3−2s )L 1 (s)L −4 (s) − L −3 (s)L 12 (s) .
For j = 4,
42s
(1 − 2−s )L 1 (s)L −4 (s) + L −8 (s)L 8 (s) ,
2
S(0, 2, 4) = 22s+1 (1 − 2−s )L 1 (s)L −4 (s),
S(0, 1, 4) =
S(1, 1, 4) = 23s (1 − 2−s )L 1 (s)L −4 (s),
S(1, 2, 4) =
42s
(1 − 2−s )L 1 (s)L −4 (s) − L −8 (s)L 8 (s) .
2
For j = 6,
1
(−3 + 21+s − 21+2s − 32s )L 1 (s)L −4 (s)
2
− 32s L −3 (s)L 12 (s) + 62s Q(1, 0, 36) ,
1
(2s − 1)(32s − 1)L 1 (s)L −4 (s) + 32s (2s + 1)L −3 (s)L 12 (s) ,
S(1, 1, 6) =
2
1 2s
(3 − 1)L 1 (s)L −4 (s) + 32s L −3 (s)L 12 (s) ,
S(0, 2, 6) =
2
1 s s
2 (2 + 1)(32s − 1)L 1 (s)L −4 (s) − 18s (2s + 1)L −3 (s)L 12 (s) ,
S(1, 2, 6) =
4
1 2s
(3 − 1)L 1 (s)L −4 (s) − 32s L −3 (s)L 12 (s) ,
S(2, 2, 6) =
2
S(0, 3, 6) = 21+s (2s − 1)L 1 (s)L −4 (s),
1 s
(2 − 1)(32s − 1)L 1 (s)L −4 (s) − 32s (2s + 1)L −3 (s)L 12 (s) ,
S(1, 3, 6) =
2
1
S(2, 3, 6) = (−3 + 21+s − 21+2s − 32s )L 1 (s)L −4 (s)
2
− 32s L −3 (s)L 12 (s) + 62s Q(4, 0, 9) ,
S(0, 1, 6) =
S(3, 3, 6) = 4(2s − 1)L 1 (s)L −4 (s).
168
Use of Dirichlet series with complex characters
It may also be seen that the sum of two unknown members may also be
expressed in terms of Dirichlet series; thus
S(0, 1, 6) + S(2, 3, 6) = 2s−1 (2s − 1)(32s − 1)L 1 (s)L −4 (s)
+ 2s−1 32s (2s + 1)L −3 (s)L 12 (s).
Whether Q(1, 0, 36) or Q(4, 0, 9) may be individually expressed in terms of
Dirichlet series with complex characters is still an open question.
For j = 5, the previous method can be used to establish one factorization, see
[15]:
S(1, 2, 5) = (5s − 1)L 1 (s)L −4 (s).
The other results for j = 5 all involve Dirichlet series with complex characters
and have been obtained, see McPhedran et al. [11], by evaluating coefficients
∞
−s for a sufficient set of c . These
cn in the expansion S(k, l, 5) =
n
n=1 cn n
may be compared with the corresponding expansions generated from appropriate
combinations of Dirichlet functions, where, if a product with a pair of complex
characters occurs, this must be accompanied by the product with complex conjugated characters, to ensure a real result for Im s = 0. The following two complex
L-functions of order 20 were required:
L i20 (s) = (20, 1)+ + i(20, 3)+ − i(20, 7)+ − (20, 9)+ ,
L −i
20 (s) = (20, 1)+ − i(20, 3)+ + i(20, 7)+ − (20, 9)+ .
Then, defining
a5(s) = (1 − 5−s )2 L 1 (s)L −4 (s) − L 5 (s)L −20 (s),
b5(s) = (1 − 5−s )2 L 1 (s)L −4 (s) + L 5 (s)L −20 (s),
−i
c51(s) = L i−5 (s)L i20 (s) + L −i
−5 (s)L 20 (s),
−i
(s)L
(s)
,
c52(s) = i L i−5 (s)L i20 (s) − L −i
−5
20
we have the solutions
52s
[b5(s) + c51(s)],
4
52s
S(1, 1, 5) =
[a5(s) − c52(s)],
4
52s
S(0, 2, 5) =
[b5(s) − c51(s)],
4
52s
S(2, 2, 5) =
[a5(s) + c52(s)].
4
S(0, 1, 5) =
Note that, in the expansion of S( p, r, j) as a sum over factors 1/n s , all terms
with non-zero coefficients have n ≡ ( p 2 + r 2 ) mod j. For j = 5, each modulus
4.4 Expressions for displaced lattice sums in closed form for j = 2−10 169
value 1, 2, 3, 4, or 5 occurs only once. We believe the results given above are the
first examples of the expression of real lattice sums in terms of real and complex
L-series. Alternating forms of these sums can also be found; for example,
∞
(−1)m+n
m,n=−∞
(5m)2 + (5n + 1)2
s
=
1
(1 − 5−s )2 (1 − 21−s )L 1 (s)L −4 (s)
4
+ (1 + 21−s )L 5 (s)L −20 (s)
+ (1 − i21−s )L i20 (s)L i−5 (s)
−i
+ (1 + i21−s )L −i
(s)L
(s)
.
20
−5
(4.4.9)
1−s )L (s) → log 2. It is simple
The two sides converge for s = 1. As s → 1, (1−2
1 √
√
to show that L −4 (1) = π/4, L 5 (1) = 2 log ( 5 + 1)/2 / 5, and L −20 (1) =
√
π/ 5. The evaluation of L i−5 (1) and L i20 (1) is somewhat more difficult. We find
√
2 π 3 + 4i 1/4
L i−5 (1) =
,
5
5
and, with the aid of Mathematica,
L i20 (1)
'
'
√
√
√
1
=− √
5 − 5 log −1 + 5 − 5 − 2 5
5 2
'
'
√
√
√
+ 5 + 5 log 1 + 5 − 5 + 2 5
'
'
√
√
√
− i 5 + 5 log −1 + 5 + 5 − 2 5
'
'
√
√
√
+ 5 − 5 log 1 + 5 − 5 + 2 5 .
−i
Similar results for L −i
−5 (1) and L 20 (1) lead to
∞
√
(−1)m+n
π
log(11 + 5 5)
=
2
2
25
(5m) + (5n + 1)
m,n=−∞
'
√
√
√ − 5 log 5 + 1 − 5 + 2 5 . (4.4.10)
In the same way, the following results may also be obtained.
√
∞
(11 + 5 5)
(−1)m+n
π
log
=
25
4
(5m + 1)2 + (5n + 1)2
m,n=−∞
√
'
√
√
5π
+
log
5−1+ 5−2 5 ,
25
170
Use of Dirichlet series with complex characters
∞
√
(−1)m+n
π
log(11
+
5
=
−
5)
25
(5m)2 + (5n + 2)2
m,n=−∞
√
'
√
√
5π
log
−
5+1− 5+2 5 ,
25
√
∞
m+n
(−1)
π
(11 + 5 5)
=
log
25
4
(5m + 2)2 + (5n + 2)2
m,n=−∞
√
'
√
√
5π
−
log
5−1+ 5−2 5 ,
25
√
∞
m+n
(−1)
5+1
π
=
log 2 − log
2
2
5
2
m,n=−∞ m + (5n + 1)
√
'
√
√
5
log −19+9 5+3 85−38 5 ,
+
5
and
∞
(−1)m+n
π
=−
2
2
5
m,n=−∞ m + (5n + 2)
√
log 2 − log
5+1
2
√
'
√
√
5
−
log −19 + 9 5 + 3 85 − 38 5 .
5
This illustrates that, since the sum is real, combinations of complex L-series series
will lead to a real number, which in this case could be expressed in elementary
constants of analysis.
Equation (4.4.10) was submitted as a problem to the American Mathematical
Monthly by Zucker and McPhedran [16]. It stimulated Berndt et al. [3] to seek
alternative methods to solve similar problems. They considered
F(a,b) (x) :=
∞
(−1)m+n
,
(xm)2 + (an + b)2
m,n=−∞
and showed it was possible to evaluate the latter for any positive rational value of
x and for many values of (a, b) ∈ N2 . They proved that
F(a,b) (x) = −
a−1
2π −(2 j−1)b
ω
ax
j=0
log
∞
(1 − ω2 j+1 q 2m+1 )(1 − ω−2 j−1 q 2m+1 ) ,
m=0
where
ω = eπi/a
and
q = e−π/x .
(4.4.11)
4.4 Expressions for displaced lattice sums in closed form for j = 2−10 171
Then, using elliptic functions, singular moduli, class invariants, and the Rogers–
Ramanujan continued fraction, they solved (4.4.11) for a ∈ {3, 4, 5, 6} and for a
variety of examples, producing such specimens as
√ (−1)m+n
π
log
8(4
−
=
15) ,
√
m 2 + 5(3n + 1)2
9 5
m,n=−∞
1/8
∞
2 +1
(−1)m+n
π
,
= √ log 1/8
(8m)2 + (4n + 1)2
2 −1
8 2
m,n=−∞
∞
as well as the problem sum (4.4.11). For larger even values of a another formula
for F(a,b) (x) in terms of the Jacobian elliptic function dn was given, and an evaluation of F10,1 (1) was carried out. The methods of Berndt et al. [3] seem to be
completely disjoint from the procedures employed here.
Following the method described above for j = 5 it has been possible to find
similar results for larger values of j. For j = 7, there are nine independent lattice sums. Of these three on their own have been solved. Pairs of the other six
sums could also be evaluated in terms of real and complex L-series. New complex
L-series of order 28 were required, and these were
(28, 1)− + ω(28, 3)− + ω2 (28, 5)− + ω2 (28, 9)− + ω(28, 11)− + (28, 13)−
= L ω−28 ,
(28, 1)− − ω2 (28, 3)− − ω(28, 5)− − ω(28, 9)− − ω2 (28, 11)− + (28, 13)−
= L −ω
−28 ,
2
(28, 1)+ − ω(28, 3)+ − ω2 (28, 5)+ + ω2 (28, 9)+ + ω(28, 11)+ − (28, 13)+
= L −ω
28 ,
(28, 1)+ + ω2 (28, 3)+ + ω(28, 5)+ − ω(28, 9)+ − ω2 (28, 11)+ − (28, 13)+
= L ω28 .
2
We let
a7(s) = (1 − 7−2s )L 1 (s)L −4 (s) − L −7 (s)L 28 (s),
b7(s) = (1 − 7−2s )L 1 (s)L −4 (s) + L −7 (s)L 28 (s),
−ω
c71(s) = L ω7 (s)L ω−28 (s) + L −ω
7 (s)L −28 (s),
2
2
−ω
ω
c72(s) = L ω−7 (s)L −ω
28 (s) + L −7 (s)L −28 (s),
2
2
−ω
c73(s) = ωL ω7 (s)L ω−28 (s) − ω2 L −ω
7 (s)L −28 (s),
2
2
2 −ω
ω
c74(s) = ωL ω−7 (s)L −ω
28 (s) − ω L −7 (s)L −28 (s),
2
2
ω2 (s) ω
−ω2
L −28 (s) − ωL −ω
7 (s)L −28 (s),
c75(s) = ω2 L 7
−ω
ω
c76(s) = ω2 L ω−7 (s)L −ω
28 (s) − ωL −7 (s)L −28 (s).
2
2
172
Use of Dirichlet series with complex characters
Then
72s
[a7(s) + c71(s) − c72(s)] ,
12
72s
[a7(s) − c73(s) + c74(s)] ,
S(1, 2, 7) =
12
72s
[a7(s) + c75(s) − c76(s)] .
S(1, 3, 7) =
12
S(2, 3, 7) =
These three sums, which have been solved, have expansions involving terms with
values 6, 5, and 3 (mod 7) respectively:
72s
[b7(s) + c71(s) + c72(s)] ,
6
72s
[b7(s) − c73(s) − c74(s)] ,
S(0, 3, 7) + S(1, 1, 7) =
6
72s
[b7(s) + c75(s) + c76(s)] .
S(0, 2, 7) + S(3, 3, 7) =
6
S(0, 1, 7) + S(2, 2, 7) =
The three sets of pair relations above contain terms with values 1, 2, and 4 (mod 7)
respectively.
For j = 8, let
a8(s) = (1 − 2−s )L 1 (s)L −4 (s) − L −8 (s)L 8 (s),
b8(s) = (1 − 2−s )L 1 (s)L −4 (s) + L −8 (s)L 8 (s),
−i
i
i
c81(s) = L −i
−16 (s)L 16 (s) + L −16 (s)L 16 (s),
−i
i
i
(s)L
(s)
−
L
(s)L
(s)
.
c82(s) = i L −i
16
−16
−16
16
The following three individual sums have been found:
S(1, 2, 8) = 26s−3 [a8(s) + c82(s)] ,
S(1, 3, 8) = 25s−2 a8(s),
S(2, 3, 8) = 26s−3 [a8(s) − c82(s)] .
The six remaining independent sums occur as three pairs:
S(0, 1, 8) + S(1, 4, 8) = 26s−2 [b8(s) + c81(s)] ,
S(0, 3, 8) + S(3, 4, 8) = 26s−2 [b8(s) − c81(s)] ,
S(1, 1, 8) + S(3, 3, 8) = 25s−1 b8(s).
4.4 Expressions for displaced lattice sums in closed form for j = 2−10 173
For j = 9 there are 12 independent terms; the following L-series of order 36
were required:
(36, 1)− − ω(36, 5)− − ω2 (36, 7)− − ω2 (36, 11)− − ω(36, 13)− + (36, 17)−
= L −ω
−36 ,
(36, 1)− + ω2 (36, 5)− + ω(36, 7)− + ω(36, 11)− + ω2 (36, 13)− + (36, 17)−
= L ω−36 ,
2
(36, 1)+ + ω(36, 5)− − ω2 (36, 7)+ + ω2 (36, 11)+ − ω(36, 13)+ − (36, 17)+
= L ω36 ,
(36, 1)+ − ω2 (36, 5)− + ω(36, 7)+ − ω(36, 11)+ + ω2 (36, 13)+ − (36, 17)+
= L −ω
36 .
2
Let
a9(s) = (1 − 3−2s )L 1 (s)L −4 (s) − L −3 (s)L 12 (s),
b9(s) = (1 − 3−2s )L 1 (s)L −4 (s) + L −3 (s)L 12 (s),
−ω
ω
c91(s) = L ω9 (s)L −ω
−36 (s) + L 9 (s)L −36 (s),
2
2
−ω
ω
ω
c92(s) = L −ω
−9 (s)L 36 (s) + L −9 (s)L 36 (s),
2
2
2 −ω
ω
c93(s) = ωL ω9 (s)L −ω
−36 (s) − ω L 9 (s)L −36 (s),
2
2
−ω
ω
2 ω
c94(s) = ωL −ω
−9 (s)L 36 (s) − ω L −9 (s)L 36 (s),
2
2
−ω
ω
c95(s) = ω2 L ω9 (s)L −ω
−36 (s) − ωL 9 (s)L −36 (s),
2
2
−ω
ω
ω
c96(s) = ω2 L −ω
−9 (s)L 36 (s) − ωL −9 (s)L 36 (s).
2
2
The 12 sums divide into six pairs, giving
S(2, 2, 9) + 2S(1, 4, 9) =
S(1, 1, 9) + 2S(2, 4, 9) =
S(4, 4, 9) + 2S(1, 2, 9) =
S(0, 1, 9) + 2S(1, 3, 9) =
S(0, 4, 9) + 2S(3, 4, 9) =
S(0, 2, 9) + 2S(2, 3, 9) =
92s
6
92s
6
92s
6
92s
6
92s
6
92s
6
[a9(s) + c91(s) − c92(s)] ,
[a9(s) − c93(s) + c94(s)] ,
[a9(s) + c95(s) − c96(s)] ,
[b9(s) + c91(s) + c92(s)] ,
[b9(s) − c93(s) − c94(s)] ,
[b9(s) + c95(s) + c96(s)] .
174
Use of Dirichlet series with complex characters
These results as displayed are associated with terms having the values
8, 2, 5, 1, 7, 4 (mod 9) respectively. The similarity between these results and those
for j = 7 is noteworthy.
For j = 10 there are 13 independent terms and the complex L-series of order
20, given for j = 5, were required. Let
a10(s) = (1 − 2−s )(1 − 5−s )2 L 1 (s)L −4 (s) − (1 + 2−s )L 5 (s)L −20 (s),
b10(s) = (1 − 2−s )(1 − 5−s )2 L 1 (s)L −4 (s) + (1 + 2−s )L 5 (s)L −20 (s),
−i
c10(s) = L i−5 (s)L i20 (s) + L −i
−5 (s)L 20 (s),
−i
d10(s) = i L i−5 (s)L i20 (s) − L −i
−5 (s)L 20 (s) .
Seven of the 13 independent terms could be solved exactly:
102s
b10(s) + c10(s) − 2−s d10(s) ,
2s+2
S(1, 3, 10) = 10s (1 − 2−s )(1 − 5−s )L 1 (s)L −4 (s),
S(1, 1, 10) =
S(1, 4, 10) =
S(1, 5, 10) =
S(2, 3, 10) =
S(3, 3, 10) =
S(3, 5, 10) =
102s
8
102s
2s+2
102s
8
102s
2s+2
102s
2s+2
a10(s) − 2−s c10(s) − d10(s) ,
a10(s) + 2−s c10(s) + d10(s) ,
a10(s) + 2−s c10(s) + d10(s) ,
b10(s) − c10(s) + 2−s d10(s) ,
a10(s) − 2−s c10(s) − d10(s) .
The remaining six form three pairs which have solutions:
102s
b10(s) + c10(s) − 2−s d10(s) ,
4
102s
S(0, 3, 10) + S(2, 5, 10) =
b10(s) − c10(s) + 2−s d10(s) ,
4
102s
S(1, 2, 10) + S(3, 4, 10) = s (1 − 2−s )(1 − 5−s )L 1 (s)L −4 (s).
5
S(0, 1, 10) + S(4, 5, 10) =
It appears to us that there is no apparent reason why similar results cannot be
found for larger values of j. However, at this stage we have no rules for determining which particular S( p, r, j) or combination of such terms can be put into
closed form, and some criteria are desirable in order to go further.
4.5 Exact solutions of lattice sums involving indefinite quadratic forms 175
4.5 Exact solutions of lattice sums involving indefinite
quadratic forms
Efforts to solve Q(a, b, c) (see (4.2.10)) in terms of L-series have concentrated on
the case when the binary quadratic form am 2 +bmn +cn 2 is positive definite, i.e.,
a > 0 and the discriminant b2 − 4ac < 0. Following Lorenz [9], who found an
exact form for T (1, 0−1), defined below, Zucker and Robertson [18] attempted to
solve certain lattice sums involving indefinite quadratic forms. They investigated
| p 2 m 2 − r 2 n 2 |−s = T ( p 2 , 0, −r 2 ; s) := T ( p 2 , 0, −r 2 ), (4.5.1)
p 2 m 2 =r 2 n 2
and in particular found an expression for T (1, 0, −r 2 ) in terms of (k, l) symbols.
This was
r −1
2
(2r, t) + (2r, 2r − t) .
T (1, 0, −r 2 ) = 4r −2s 1 − 21−s + 21−2s L 21 + 2
t=1
(4.5.2)
For r = 1−6 it was possible to find solutions in terms of the squares of positiveparity real L-series, but for larger values of r no such solutions could be found.
Now, it was noted in Section 4.3 that every (k, l) is expressible as a linear combination of L-series of period k if the complex L-series are included. Thus in
principle (4.5.2) may be written in L-series for every r . To illustrate this, closed
forms for r = 1 − 13 have been evaluated; the outcome is displayed in Table 4.1.
In this table we have
1
T (1, 0, −r 2 ) =
2 − r 2 n 2 |s
|m
2
2 2
m =r n
and
ω = exp
iπ
3
,
τ = exp
iπ
5
,
ρ = exp
iπ
6
.
These results show that in addition to squares of positive-parity L-series with
real characters, products of pairs of positive-parity complex L-series will in general be required in order to solve T (1, 0, −r 2 ). It is apparent that with sufficient
labour there is no limit on how far one may go, but until now no clear pattern has
yet emerged.
We have established numerically, but not analytically, the following functional
equation:
sπ π 2s−1 (1 − s)
?
tan
.
T (1, 0, −r 2 ; s) = T (1, 0, −r 2 ; 1 − s)
r
(s)
2
It may also be shown that T has the following expansion near its second-order
pole at s = 1:
2 2
2
4γ
+
γ + 2γ1 + C(r ) + O[s].
T (1, 0, −r 2 ; 1 + s) = 2 +
rs
r
rs
176
Use of Dirichlet series with complex characters
Table 4.1 The functions T (1, 0, −r 2 ) for r = 1−13
r
1
2
3
4
5
6
7
8
9
10
11
12
13
T (1, 0, −r 2 )
4 1 − 21−s + 21−2s L 21 (s)
√
√
2 1+
2 − 1 2−s + 21−2s 1 −
2 + 1 2−s + 21−2s L 21 (s)
2 1 − 21−s + 21−2s 1 − 2 × 3−s + 31−2s L 21 (s)
1 − 2−s + 21−2s 1 − 2−s − 21−3s + 22−4s L 21 (s) + L 28 (s)
1 − 21−s + 21−2s 1 − 2 × 5−s + 51−2s L 21 (s) + 1 + 21s + 21−2s L 25 (s)
√
√
1+
2 − 1 2−s + 21−2s 1 −
2 + 1 2s + 21−2s
× 1 − 2 × 3−s + 31−2s L 21 (s) + L 212 (s)
2 1 − 21−s + 21−2s
ω2
1 − 2 × 7−s + 71−2s L 21 (s) + 43 1 + 2s + 21−2s L −ω
7 (s)L 7 (s)
3
1 1 − 21−s +3 × 2−2s − 22−3s +3 × 21−4s − 23−5s + 3 × 22−6s − 24−7s + 24−8s L 2 (s)
1
2
+ 12 (1 + 21−2s )L 28 (s) + L i16 (s)L −i
16 (s)
√
√
2 1 − 21−s + 21−2s
1+
3 − 1 3−s + 31−2s 1 −
3 + 1 3−s + 31−2s L 21 (s)
3
ω2
+ 43 1 + 2−s + 21−2s L −ω
9 (s)L 9 (s)
√
√
1 1+
2 − 1 2−s + 21−2s 1 −
2 + 1 2−s + 21−2s 1 − 2 × 5−s + 51−2s L 21 (s)
2
√
√
+ 12 1 + − 2 + 1 2−s + 21−2s 1 +
2 + 1 2−s + 21−2s L 25 (s) + L i20 (s)L −i
20 (s)
√ −1−s
2 1 − 21−s + 21−2s
4
2
−s
1−2s
+ 21−2s
1 − 2 × 11 + 11
L 1 (s) + 5 1 + (1 − 5)2
5
√ −1−s
3
2
4
τ4
+ 21−2s L −τ
× L τ11 (s)L −τ
11 (s)L 11 (s)
11 (s) + 5 1 + (1 + 5)2
1 1 − 2 × 3−s + 31−2s
1 − 2−s + 21−2s 1 − 2s − 21−3s + 22−4s L 21 (s)
2
+ 12 1 + 2 × 3−s + 31−2s L 28 (s) + 12 1 + 21−2s L 212 (s) + 12 L 224 (s)
1 1 − 21−s + 21−2s
1 − 2 × 13−s + 131−2s L 21 (s) + 13 1 + 2−1−s + 21−2s L 213 (s)
3
2
ρ
−ρ 4
−ρ 2
ρ4
+ 23 1 − 2−s + 21−2s L 13 (s)L 13 (s) + 23 1 + 2s + 21−2s L 13 (s)L 13 (s)
The quantity γ1 is the first Stieltjes constant (which occurs in the Laurent series
expansion of the Hurwitz zeta function), and C(r ) is a constant depending on the
integer r . Its first three values are
C(1) = 2 log2 2,
C(2) =
3
log2 2,
2
C(3) =
2
1
log2 2 + log2 3.
3
3
The expansion near s = 0 is then
T (1, 0, −r 2 ; s) = 1 + 2 log
π r
s + O(s 2 ).
4.6 Commentary: Quadratic forms and closed forms
177
In this section it has been shown that the introduction of L-series with
complex characters has enabled closed forms to be established for many twodimensional lattice sums which previously could not be expressed in this fashion.
An obvious question is whether complex-character L-series can play a role in
higher-dimensional sums, and this may be an interesting path to pursue.
4.6 Commentary: Quadratic forms and closed forms
(1) The collaboration of M. L. Glasser and I. J. Zucker on lattice sums commenced
with their mutual interest in finding closed forms for double sums of the form
Q(a, b, c; s) =
∞
(am 2 + bmn + cn 2 )−s ,
m,n=0,0
where a > 0 and its discriminant b2 − 4ac < 0. In this case the binary
quadratic form is positive definite. In Chapter 1 we discussed how, for these
forms which had one class per genus, the double sum above could be expressed
as sums of pairs of real-character L-series of opposite parity, and the results
are displayed in Table 1.6. Altogether, 101 of these discriminants are known,
and if any others exist it is known there can be at most only one more. At
one time it was thought that these were the only Q sums expressible in this
manner. However, it was later found, in certain cases of discriminants with
two classes per genus, that if the two classes which appeared in one genus
were the related quadratic forms am 2 ± bmn + cn 2 then, since Q(a, b, c) =
Q(a, −b, c), the lattice sum involving these quadratic forms could also be
found as sums of pairs of real L-series. Several examples were given in
[18]. These turned out to be in the nature of flukes. A theorem of Chowla
[5] shows that the number of two-class-per-genus discriminants must also be
finite.
(2) In this chapter the successful use of L-series with complex characters in
expressing some two-dimensional displaced lattice sums in closed form has been
demonstrated. Further, the particular set of lattice sums involving the indefinite quadratic forms T (1, 0, −r 2 ) has also been given in closed form using
both real and complex L-series. This, as mentioned in the main part of this
chapter, has raised the hope that the latter may be applicable to other sums
involving positive definite quadratic forms. Immediate candidates for analysis would be Q(1, 0, 36) and Q(4, 0, 9), whose sum in terms real L-series is
known as is also the sum of the displaced lattice sums S(0, 1, 6) and S(2, 3, 6).
Since all other displaced lattice sums of the form S( p, q, 6) could individually be expressed in terms of real L-series, the exclusion of S(0, 1, 6) and
S(2, 3, 6) seems an anomaly. There are no period-3 or period-6 L-series with
complex characters. So if a solution is possible then it seems likely that period9 complex L-series would be involved. But whether they should be combined
178
Use of Dirichlet series with complex characters
with period-16 or period-36 complex L-series is not clear. Also, algebraic factors
such as (1 − 2−s + 21−2s ) and (1 + 31−2s ) may complicate matters even further,
so as yet no definite statement can be made about Q(1, 0, 36) and Q(4, 0, 9)
individually.
The reader should also consult Chapter 8, where lattice sums involving arbitrary
quadratic forms are analyzed.
(3) In a recent work by Guillera and Rogers [8], it has been shown that certain
sums of the form
∞
n!3
a − bn −n
z
(4.6.1)
1
(s)n ( 2 )n (1 − s)n n 3
n=1
may be evaluated as a Dirichlet L-series at 2. The summand of (4.6.1) is essentially the reciprocal of the summand of a Ramanujan series for 1/π and, as such,
sums of this form are called companion series by the authors. When a companion
series is divergent, the value (4.6.1) is given by the analytic continuation of its
hypergeometric representation.
We now very briefly outline the methods used in [8]. The idea of completing a
hypergeometric function is introduced, that is, replacing the summation index n
in a Ramanujan series by n + x, and extending the summation range to Z. The
result is a function
(s)n+x ( 1 )n+x (1 − s)n+x
2
z n+x ,
Yx (z) =
3
(1)
n+x
n∈Z
which is clearly periodic in x with period 1. This fact, combined with complex
analytic arguments – including an estimate of Yx (z) when Im x is large, allow the
authors to relate Yx (z) to the original series by some trigonometric terms.
The companion series is then extracted as a coefficient in the x-expansion of
Yx (z). The coefficients are extracted, after significant work aided by θ -function
identities, as integrals of Eisenstein series. Finally, using trigonometric partial
fractions, the companion series is expressed as the sum of two terms, each of the
form Q(a, b, c; s).
Hence, when both Q’s are solvable in terms of L-series (using Table 1.6), the
companion series (4.6.1) also simplifies in terms of L-series. However, when one
Q does not (apparently) reduce to L-series, taking a suitable hypergeometric lefthand side of the companion series allows one to rapidly compute the lattice sum.
In particular, [8] shows that Q(1, 0, 36; 2) evaluates to a linear combination of
L −4 (2), L −3 (2), and a 5 F4 with argument ≈ −4 × 10−7 , giving very fast convergence. Similar expressions for Q(1, 0, 20; 2) and Q(1, 0, 52; 2) are also given.
4.7 Commentary: More on numerical discovery
Commentaries 3.9 and 3.10 at the end of Chapter 3 describe how one can discover
a simple factorization of a lattice sum as a product of L-series, by comparing terms
4.8 Commentary: A Gaussian integer zeta function
179
in the series expansion n cn /n s of both sides, or by matching the distribution of
zeros on the critical line.
We remark that, numerically, it appears that all L-series, whether real or complex, of the same period k have the same distribution of zeros on the critical line;
namely, the number of zeros on 1/2 + it with t ∈ [0, N ] is asymptotically
N
Nk
log
−1 .
2π
2π
Here, we describe another method that can be used to discover a factorization.
As a toy example, consider equation (4.4.9). The left-hand side sum may be easily
evaluated naïvely to (say) 25 digits for s = 4 (often a suitable integral can give
many more digits). If we do not know the right-hand side, we could reasonably
guess that (for fixed s) it will be a linear combination of
(1 − 5−s )2 L 1 (s)L −4 (s),
L 5 (s)L −20 (s),
L i20 (s)L i−5 (s),
−i
L −i
20 (s)L −5 (s),
based on the factorization of S(a, b, 5) derived earlier in the chapter. (In fact,
knowing that both sides must converge when s = 1, we could argue that the
factor 1 − 21−s must also appear in front of the L 1 (S) term.) Now, each of the
four products of L-series can be easily computed to high precision, and we input
these values, as well as the approximate sum, as a vector v into an integer relation
program such as PSLQ (see e.g., [4]).
The PSLQ algorithm, which is well implemented in Maple, takes an input vector v of real or complex numbers and attempts to find a vector u of elements in
R, where R is usually the integers (but can also be the Gaussian integers), such
that v · u = 0 within the prescribed precision. In this example, even setting the
precision to as low as 20 digits (we can afford to do this because our guess has
only four terms) returns, for s = 4,
u = {7, 9, 8 − i, 8 + i, −32},
from which we readily guess that 7 = 2s−1 − 1, 9 = 2s−1 + 1, 32 = 2s+1 , etc.,
and therefore recover (4.4.9).
4.8 Commentary: A Gaussian integer zeta function
One may evaluate in closed form
1
1
ζG (N ) :=
=
zN
(m
+
in) N
m,n
Z (i)
for positive even-integer N > 1. Here, as always, the primes denote that summation avoids the pole at 0. Note that ζG (2n) has been introduced already as S2n in
Chapter 3, equation (3.1.7). The sum ζG (2n) is also a special case of an Eisenstein
series.
180
Use of Dirichlet series with complex characters
Hint: The evaluation is implicitly covered in [4, pp. 167–70] and in [10]. It relies
on analysis of the Weierstrass ℘-function; see [14, Chapter 23]. We recall that
℘ (x) :=
m,n
1
1
−
.
2
(2in + 2im − x)
(2in + 2im)2
We then differentiate twice and extract the coefficients of ζG (2n). For N divisible by 4, the sum is actually is a rational multiple of powers of the Weierstrass
invariants
1 4
1 1 4
β
=
g2 = K √
4
4
2
and g3 = 0. This is the so-called lemniscate case of ℘. Here β(x) := B(x, x) =
(x)2 / (2x) is the central beta function.
One can show that the general formula for N ≥ 1 is
ζG (4N ) =
where p1 = 1/20 and
4N
1
pN
2K √
,
4N − 1
2
N −1
pm p N −m
3 m=1
,
pN =
(4N + 1)(2N − 3)
for N > 1; compare with (3.2.5). The next three values are p2 = 1/1200, p3 =
1/156000 and p4 = 1/21216000. The corresponding values of q N := 16 N p N
are
16
128
256
4
,
,
,
.
5
75
4875
82875
Finally note that q N satisfies the same recursion as PN . This leads to the simpler
expression
qN
K
ζG (4N ) =
4N − 1
1
√
2
4N
qN
=
4N − 1
β( 14 )
4
4N
.
(4.8.1)
By contrast, it is easy to show that
ζG (2N + 1) = ζG (4N + 2) = 0.
Note that ζG is the two-dimensions analogue of the Riemann zeta function for
even arguments, since, for N > 1,
1
0,
N odd,
=
N
n
2ζ (N ), N even.
n
4.9 Commentary: Gaussian quadrature
181
Moreover, we have the closed form
ζ (2N ) =
&2N
(−1) N +1 b2N
(−1) N +1 b2N %
(2π )2N =
2β( 12 ) ,
2(2N )!
2(2N )!
(4.8.2)
where b N is the N th Bernoulli number, defined by the exponential generating
N
t
function ∞
N =0 b N t /N ! = t/(e − 1).
The generalized quantum sum is defined by
1
,
Y(a, b, c) :=
(z − a)(z − b)(z − c)
m,n=0
where z = m + ni and a, b, c are not on the lattice. One may show that Y has a
formal power series development in ζG (n), by writing the summand as
a
a2
1
b b2
c c2
1
+
+
+
+
+
·
·
·
1
+
+
·
·
·
1
+
+
·
·
·
.
z
z
z
z3
z2
z2
z2
This leads, for |a|, |b|, |c| < 1, to
Y(a, b, c) =
∞
π4n−3 (a, b, c) ζG (4n),
(4.8.3)
n=1
where
πn (a, b, c) :=
a k1 b k2 c k3 .
k1 ,k2 ,k3 ≥0
k1 +k2 +k3 =n
In particular, πn (a, a, a) = 12 (n+1)(n+2) a n . There is a corresponding expansion
for more than three linear terms.
4.9 Commentary: Gaussian quadrature
The classical theory of Gaussian quadrature is a scheme for approximating integrals by finite sums involving orthogonal polynomials (see [1, Chapter 5]). Here
we describe a new, curious, and not yet fully explored procedure to numerically
evaluate lattice sums using Gaussian quadrature.
We recall some facts about orthogonal polynomials. For a reasonably wellbehaved function ω(x), there exists a sequence of polynomials { pn (x)}∞
n=0 (where
n is also the degree), such that they are orthogonal with respect to ω:
b
pm (x) pn (x) ω(x) dx = h n δmn .
(4.9.1)
a
Here δmn is the Kronecker delta. Every polynomial of degree n can be expressed
uniquely as a linear combination of p0 , . . . , pn . We may normalize pn so that they
182
Use of Dirichlet series with complex characters
are monic, in which case it is easy to show that they satisfy the recurrence relation
pn+1 = (x − an ) pn (x) − bn pn−1 (x),
with bn = h n / h n−1 .
(4.9.2)
√
2
(For example, taking −a = b = ∞, ω(x) = e−x /2 , h n = 2π n!, an = 0, and
bn = n we get the Hermite polynomials.) With this set-up, we have
Proposition 4.1 (Gaussian quadrature) For a reasonable function f ,
b
n
f (x) ω(x) d x =
f (xi )wi + Rn ,
(4.9.3)
a
i=1
where xi are the roots of pn (x), wi are weights, defined by
wi =
−h n
,
pn+1 (xi ) pn (xi )
(4.9.4)
and Rn is the error, which depends on the (2n)th derivative of f . In particular, if
f is a polynomial of degree ≤ 2n − 1, Rn = 0 and the quadrature is exact.
By its very construction using the roots of the polynomials pn , Gaussian
quadrature is exact for polynomials of degree up to n − 1; orthogonality gives
exactness for higher degrees and it is this pleasant property which makes Gaussian quadrature superior to many other schemes. In practice, the error is difficult
to compute and is often best estimated on a case-by-case basis by fixing f and
increasing n. Heuristically, if f is closely approximated by polynomials on (a, b)
(e.g., if it has a close-fitting Taylor series) then Gaussian quadrature tends to work
well – thus we need to choose ω carefully to make f well behaved.
Engblom [7] was one of the first to give a comprehensive report on Gaussian
quadrature using discrete measures. He noted that the classical theory carried over
exactly if ω is discrete, that is, if x pm (x) pn (x)ω(x) = h n δmn ; then
x
f (x)ω(x) ≈
n
f (xi )wi .
(4.9.5)
i=1
(An example of such a class of polynomials is the Poisson–Charlier polynomials,
where ω(x) = a x /x!.)
Monien [13], using reciprocal polynomials, considered and found applications
for orthogonal conditions of the form
1 11
qm 2 qn 2 2 = h n δmn .
x
x x
x>0
He also found the recurrence and closed form (in terms of Bessel functions) for
qn (x). Quadrature using qn works particularly well if the summand f (x) admits
an asymptotic expansion in powers of 1/x 2 for large x. Here the classical theory still applies: one computes the roots xi of qn (x), from which wi follows.
4.9 Commentary: Gaussian quadrature
183
Since qn (1/x 2 ) is used instead of qn (x), the right-hand side of (4.9.5) becomes
n
√
i=1 f (1/ x i )wi .
Given some ω, it remains to compute xi and wi for quadrature to work. Note
that for fixed n we only need to calculate these numbers once – to several hundred
significant figures say, so, once set up, the cost for approximating a range of sums
is very small.
First, we generate a table of orthogonal polynomials pn using the Gram–
Schmidt orthonormalization process. We then find the roots, using for example
Newton’s method (when the table is small, e.g., less than 50 polynomials) or
exploiting the fact that the n simple zeros of pn are eigenvalues of a tridiagonal matrix, and stable and fast algorithms exist for finding them. The weights can
then be computed, either using (4.9.4) or by noting that Gaussian quadrature is
exact for low-degree elementary polynomials times ω (whose sums may be found
independently). In this way, finding wi is equivalent to inverting a Vandermonde
matrix whose columns are powers of xi .
We now demonstrate how Gaussian quadrature can be applied to lattice sums.
As a very simple example, we look at the Hardy–Lorenz sum
1
= 4β(s)ζ (s).
(4.9.6)
2 + n 2 )s
(m
m,n
Consider the s = 1 case (the argument is similar for other integers s). We can in
fact perform one summation exactly, since
∞
n=−∞
m2
1
π
= coth(mπ ),
2
m
+n
and we differentiate both sides with respect to m. The resulting sum (replacing
m by x) behaves like π/(4x 3 ) − 1/(2x 4 ) for large x, so the polynomials f n with
orthogonality conditions
∞
x=1
fn
1
x
fm
1 1
= δmn h n
x x2
(4.9.7)
can be used for quadrature. Using the Gram–Schmidt process, we obtain
f 0 (x) = 1, f 1 (x) = x − 6ζ (3)/π 2 , . . . The roots of f n (x) serve as xi , the weights
are computed using (4.9.4), and the quadrature rule becomes
∞
x=1
f (x)
n
1
1
wi .
≈
f
xi
x2
i=1
Using n = 20 (i.e., we only need to compute f 20 together with the roots and
weights), we obtain 31 correct digits for the s = 2 case of the Hardy–Lorenz sum.
Of course, for many lattice sums it is not possible to perform one summation
explicitly; also, the sums may be alternating. Our next example tackles both these
184
Use of Dirichlet series with complex characters
problems. Consider
S=
(−1)m+n
.
(m 2 + n 2 )2
m,n
For Gaussian quadrature, we use the polynomials gn with orthogonality
conditions
1 1 (−1)x
gm
= δmn h n .
gn
(4.9.8)
x
x
x
x
Gram–Schmidt gives g0 (x) = 1, g1 (x) = x − π 2 /(12 log 2), . . . We compute xi
and wi as before and use the quadrature rule
∞
x=1
f (x)
n
1
(−1)x
wi .
f
≈
x
xi
i=1
We perform a double quadrature on S, that is, we first apply quadrature to f (n) =
mn/(m 2 + n 2 )2 (since ω(x) = (−1)x /x). This gives a finite sum over functions
of m, upon which we perform quadrature as our new function. Using only n = 10,
this method gives 15 correct digits for S.
Double Gaussian quadrature using f n and gn works well for sums of the type
m,n
(±1)m (±1)n
%
,
am 2 + bmn + cn 2 + dm + en + f )s
so it can be applied to S( p, r, j; s), Q(a, b, c; s), and F(a,b) (x), studied in this
chapter, especially when other numerical methods, for instance Mellin transforms
and θ -functions, are not straightforward to apply. We normally need to split the
sum up naturally into 1 ≤ m, n ≤ ∞ and a few other similar regions. In practice,
convergence sets in even for low n though quadrature seems to be slowest when
the sum is not alternating.
Finally, note that we are not restricted to double sums. As just one example, using triple Gaussian quadrature, the polynomials gn , and n = 25, 36
significant figures for the Madelung constant have been obtained by J. Wan
(unpublished, 2012).
References
[1] G. E. Andrews, R. Askey, and R. Roy. Special Functions. Cambridge University
Press, Cambridge, 1999.
[2] R. Ayoub. An Introduction to the Analytic Theory of Numbers. American Mathematical Society, Providence, RI, 1963.
[3] B. C. Berndt, G. Lamb, and M. Rogers. Two dimensional series evaluations via the
elliptic functions of Ramanujan and Jacobi. Ramanujan Journal, 2011.
References
185
[4] J. M. Borwein and D. H. Bailey. Mathematics by Experiment: Plausible Reasoning
in the 21st Century, 2nd edition. A. K. Peters, 2008.
[5] S. Chowla. An extension of Heilbronn’s class number theorem. Quart. J. Math.
Oxford, 5:304–307, 1934.
[6] L. E. Dickson. An Introduction to the Theory of Numbers. Dover, New York, 1957.
[7] S. Engblom. Gaussian quadratures with respect to discrete measures. Uppsala
University Technical Reports, 7:1–17, 2006.
[8] J. Guillera and M. Rogers. Ramanujan series upside-down. Preprint.
[9] L. Lorenz. Bidrag tiltalienes theori. Tidsskrift Math., 1:97–114, 1871.
[10] P. McKean and V. Moll. Elliptic Curves: Function Theory, Geometry, Arithmetic.
Cambridge University Press, New York, 1997.
[11] R. C. McPhedran, L. C. Botten, N. P. Nicorovici, and I. J. Zucker. Systematic
investigation of two-dimensional static array sums. J. Math. Phys., 48:033501, 2007.
[12] R. C. McPhedran, L. C. Botten, N. P. Nicorovici, and I. J. Zucker. On the Riemann
property of angular lattice sums and the one-dimensional limit of two-dimensional
lattice sums. Proc. Roy. Soc. A, 464:3327–3352, 2008.
[13] H. Monien. Gaussian quadrature for sums: a rapidly convergent summation scheme.
Math. Comp., 79:857–869, 2010.
[14] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (eds.). NIST Handbook
of Mathematical Functions. Cambridge University Press, New York, 2010.
[15] I. J. Zucker. Further relations amongst infinite series and products II. The evaluation
of 3-dimensional lattice sums. J. Phys. A, 23:117–132, 1990.
[16] I. J. Zucker and R. C. McPhedran. Problem 11294. Amer. Math. Monthly, 114:452,
2007.
[17] I. J. Zucker and M. M. Robertson. Some properties of Dirichlet L-series. J. Phys. A,
9:1207–1214, 1976.
[18] I. J. Zucker and M. M. Robertson. Further aspects of the evaluation of
2
2 −s
(m,n=0,0) (am + bmn + cn ) . Math. Proc. Camb. Phil. Soc., 95:5–13, 1984.
5
Lattice sums and Ramanujan’s
modular equations
A modular equation of order n is essentially some algebraic relation between
theta functions of arguments q and q n respectively. In his notebooks Ramanujan
gave many such relations involving Lambert series, and Berndt [1] painstakingly
collected these results, proved or verified them, and if necessary made corrections.
However, these relations as given by Ramanujan appear in a haphazard way, and
there seems to be no systematic way of ordering them or for that matter knowing whether the formulae are independent or complete. Here an attempt is made
to arrange these results in a systematic fashion. It will also be demonstrated how
new modular relations may be derived from those previously established. Indeed
it will be shown how, using sign and Poisson transformations to be described
in Chapter 6, each modular equation is essentially a set of either four or eight
relations which can be generated from any one of the set by a group of simple
transformations. Further it will also be shown how the use of character notation provides a shorthand for the lengthy Lambert series involved. A connection
between binary quadratic forms and modular equations will be demonstrated.
This will allow very simple proofs of certain modular and mixed modular equations to be executed. Finally it will be exhibited how the Mellin transforms of
modular equations, in which q is replaced by e−t , lead to the evaluation of
lattice sums.
All the modular equations are given in terms of the theta functions described in
Section 1.3, namely
θ2 =
∞
q
(n−1/2)2
,
θ3 =
−∞
∞
n2
q ,
θ4 =
−∞
θ5 = 2
∞
−∞
(−1)n q (2n−1/2) .
−∞
∞
2
2
(−1)n q n ,
Lattice sums and Ramanujan’s modular equations
187
Ramanujan has his own notation, and the following ‘dictionary’ applies:
Ramanujan
classical
φ(q)
θ3
ψ(q 2 )
θ2 /2q 1/4
φ(−q)
θ4
ψ(−q 2 )
θ5 /2q 1/4
The main aspects of character notation will now be described. Characters were
introduced in Section 1.4 in the definition of Dirichlet L-series. Only real characters are considered here. Primitive characters are defined as follows. If p is a
prime > 2 then
χ p (n) = χ p = (n| p),
where (n| p) is the Legendre–Jacobi–Kronecker symbol, which is ±1. The suffix
p will be signed according to whether χ p ( p − 1) = ±1. The simplest possible
character is χ 1 (n) = 1. Other examples are as follows:
χ −3 (n) = +1,
n ≡ 1 (mod 3),
= −1,
n ≡ 2 (mod 3),
= 0,
n ≡ 0 (mod 3),
χ 5 (n) = +1,
n ≡ 1, 4 (mod 5),
= −1,
n ≡ 2, 3 (mod 5),
= 0,
n ≡ 0 (mod 5).
In addition to these there are three other primitive characters. They are
χ −4 = +1,
n ≡ 1 (mod 4),
= −1,
n ≡ 3 (mod 4),
= 0,
(n, 4) = 1,
and
χ −8 (n) = +1,
n ≡ 1, 3 (mod 8),
= −1,
n ≡ 5, 7 (mod 8),
= 0,
(n, 8) = 1,
χ 8 (n) = +1,
n ≡ 1, 7 (mod 8),
= −1,
n ≡ 3, 5 (mod 8),
= 0,
(n, 8) = 1.
Further primitive characters may be formed from square-free products of
prime characters, and from prime characters multiplied by χ −4 , χ −8 , and χ 8 .
Thus χ P = χ p1 χ p2 χ p3 · · · and χ −4 χ P , χ −8 χ P , and χ 8 χ P , are all primitive
characters. The character series
L P = L P (s) =
∞
χ P (n)
n=1
ns
188
Lattice sums and Ramanujan’s modular equations
each yield an independent Dirichlet L-function. For example,
L 1 (s) =
∞
χ 1 (n)
n=1
ns
= 1−s + 2−s + 3−s + · · · = ζ (s)
(Riemann zeta function),
L −3 (s) =
∞
χ −3 (n)
n=1
ns
= 1−s − 2−s + 4−s − 5−s + · · · .
(5.0.1)
(5.0.2)
Two other types of character are encountered. These are as follows:
χ ka (n) = +1
if (k, n) = 1,
= 0
if (k, n) = 1;
χ kb (n) = +1
= −(k − 1)
if (k, n) = 1,
if n ≡ 0 (mod k).
(5.0.3)
(5.0.4)
These latter non-primitive characters are essentially modifications of the principal character, χ 1 (n) = +1 for all n. When these non-primitive characters multiply
primitive characters the latter are modified in such a way that the character series
produced are L-series of the primitive character, multiplied by a factor. Thus it
may be shown that
∞
χ P χ ka
n=1
∞
n=1
ns
χ P χ kb
ns
= 1 − (k|P)k −s L P ,
(5.0.5)
= 1 − (k|P)k 1−s L P .
(5.0.6)
Of these non-primitive characters, two occur often in this combination, namely
χ 2a and χ 2b , and when they multiply primitive characters the combination may
be represented by χ 2Pa and χ 2Pb .
Two such modified characters appear often here in connection with modular
equations of order 3. They are χ −3 χ 2a = χ −6a and χ −3 χ 2b = χ −6b , whose
character tables are
χ −6a (n) = +1,
n ≡ 1 (mod 6),
= −1,
n ≡ 5 (mod 6),
= 0,
χ −6b (n) = +1,
n ≡ 0 (mod 3),
n ≡ 1, 2 (mod 6),
= −1,
n ≡ 4, 5 (mod 6),
= 0,
n ≡ 0 (mod 3).
Lattice sums and Ramanujan’s modular equations
189
Then it is found that
∞
χ −6a
n=1
ns
−s
= (1 + 2 )L −3 ,
∞
χ −6b
n=1
ns
= (1 + 21−s )L −3 .
The values of characters for the χ ±n which appear here have been given, for all
n up to 30, in Table A.2 in the Appendix. The period of the character is denoted
by a vertical bar, | .
A modular equation will be said to have dimension d if the power of the theta
function term is d. Thus expressions such as θ (q)θ (q 3 ) or θ 3 (q)/θ (q 3 ) both have
dimension 2. The reason for describing such expressions in this way is, as will be
seen later, that the Mellin transforms of these forms lead to multiple sums over
d-dimensional lattices. Ramanujan’s modular equations are generally given in the
form of a relation between theta functions and Lambert series, where Lambert
series have the general form
∞
f (n)q mn+k
,
1 ± q r (mn+k)
n=1
with m, k, and r all integers.
To illustrate the use of character notation and the various transformations possible, two pairs of modular equations of order 3, one pair of dimension 2 and the
other of dimension 4 will be considered. They are the pair of equations numbered
(3.i) and (3.ii) and the pair (3.iii) and (3.iv) in Chapter 19 – referred to henceforth
as C.19 – of Ramanujan’s Notebooks Part III [1]:
q
q5
q7
q 11
−
+
−
+
·
·
·
,
4qψ(q 2 )ψ(q 6 ) = 4
1 − q2
1 − q 10
1 − q 14
1 − q 22
(5.0.7)
q2
q
q4
q5
−
+
−
+
·
·
·
φ(q)φ(q 3 ) = 1 + 2
1−q
1 + q2
1 + q4
1 − q5
(5.0.8)
and
q
2q 2
4q 4
5q 5
+
+
+
+ ··· ,
(5.0.9)
1 − q2
1 − q4
1 − q8
1 − q 10
4q 4
q
5q 5
7q 7
+
φ 2 (q)φ 2 (q 3 ) = 1 + 4
+
+
+
·
·
·
.
1−q
1 − q7
1 − q4
1 − q5
(5.0.10)
qψ 2 (q)ψ 2 (q 3 ) =
Consider the first pair. It is immediately apparent that these expressions are
concisely expressed in character notation, and replacing Ramanujan’s functions
with the classical notation we have
190
Lattice sums and Ramanujan’s modular equations
θ2 θ2 (q ) = 4
3
∞
χ −6a q n
n=1
θ3 θ3 (q 3 ) = 1 + 2
,
1 − q 2n
(5.0.11)
∞
χ −3 q n
n=1
1 + (−q)n
.
(5.0.12)
Now if we apply the sign transform to these expressions, a further two equations
may be obtained. It is simple to do this to (5.0.8), with the following result:
θ4 θ4 (q 3 ) − 1 = −2
∞
χ −6b q n
n=1
1 + qn
,
(5.0.13)
which is exactly what would be obtained by a Poisson transform of (5.0.7). The
sign transform of (5.0.7) requires care as q has to be replaced by iq. This yields
∞
χ 12 q n
q n − q 5n
=
4
.
1 + q 6n
1 + q 2n
n=1
n=1
(5.0.14)
These results thus form a set of four elements, all of which can be obtained by
knowing just one of them; they are connected to one another as shown in the
following scheme:
4qψ(−q 2 )ψ(−q 6 ) = θ5 (q)θ5 (q 3 ) = 4
∞
χ −4
[θ3 θ3 (q 3 ) − 1] ⇐ S ⇒ [θ4 θ4 (q 3 ) − 1] ⇐ P ⇒ θ2 θ2 (q 3 ) ⇐ S ⇒ θ5 θ5 (q 3 ),
(5.0.15)
where S and P are respectively the sign and Poisson operations.
The Mellin transform also allows us to find relations amongst theta functions
in a particularly simple fashion. Thus we have
M[θ3 θ3 (q 3 ) − 1] = 2(1 + 21−2s )L 1 L −3 ,
(5.0.16)
M[θ2 θ2 (q 3 )] = 4(1 − 2−2s )L 1 L −3 ,
M[θ4 θ4 (q ) − 1] = −2(1 − 2
3
2−2s
(5.0.17)
)L 1 L −3 .
(5.0.18)
Hence
M[θ3 θ3 (q 3 )] = M[θ2 θ2 (q 3 ) + θ4 θ4 (q 3 )]
(5.0.19)
θ3 θ3 (q 3 ) = θ2 θ2 (q 3 ) + θ4 θ4 (q 3 ),
(5.0.20)
and
which is Legendre’s classical result for the third-order modular relation.
It is pertinent to find the Mellin transform of θ3 θ3 (q 3 ) − 1 directly from the
definition of θ3 . Since
θ3 θ3 (q 3 ) − 1 =
∞ ∞
−∞ −∞
qn
2 +3m 2
−1=
qn
2 +3m 2
,
(5.0.21)
Lattice sums and Ramanujan’s modular equations
we have
M θ3 θ3 (q 3 ) − 1 =
(n 2 + 3m 2 )−s = 2(1 + 21−2s )L 1 L −3 ,
191
(5.0.22)
where implies summation over all integer values of n and m but excluding the
case when they are simultaneously zero. The reason for referring to such forms as
θ3 θ3 (q 3 ) as two-dimensional may now be seen – namely, their Mellin transform
yields a double sum over a two-dimensional lattice.
In a similar fashion (5.0.9) and (5.0.10) may be treated in exactly the same way
and from them two further modular forms of order 3 may be formed. Only four
elements are generated from each of (5.0.7) and (5.0.9). The reason is that these
elements exhibit a certain amount of symmetry, and so the S and P operations
do not yield new results. Thus for example the Poisson transform of (5.0.8) yields
itself. These eight results found from (5.0.7)–(5.0.10) are given in Table A.1 in
the Appendix.
Now, however, consider the following entries in C.19 (4.i)–(4.iv):
qψ 5 (q)ψ(q 3 ) − 9q 2 ψ(q)ψ 5 (q 3 ) =
q
22 q 2
42 q 4
−
+
1 − q2
1 − q4
1 − q8
52 q 5
−
+ ··· ,
(5.0.23)
1 − q 10
22 q 2
q
42 q 4
−
9φ(q)φ 5 (q 3 ) − φ 5 (q)φ(q 3 ) = 8 1 +
+
2
1+q
1−q
1 − q4
52 q 5
−
+ ··· ,
(5.0.24)
1 + q5
q5
q
ψ 3 (q)
q7
q 11
−
=
1
+
3
+
−
+
·
·
·
,
1−q
1 − q7
ψ(q 3 )
1 − q5
1 − q 11
q2
φ 3 (q)
q
q4
q5
+
=1+6
−
−
+ ··· .
1−q
φ(q 3 )
1 + q2
1 + q4
1 − q5
(5.0.25)
(5.0.26)
Equations (5.0.23) and (5.0.24) are third-order modular equations of dimension 6 and (5.0.25) and (5.0.26) are third-order modular equations of dimension
2. Let us consider the latter. First, we put them into classical and character
notation:
∞
χ −6a (n)q n
1 θ2 3 (q 1/2 )
=
1
+
3
,
4 θ2 (q 3/2 )
1 − qn
n=1
∞
χ −6b (n)q n
θ3 3
=
1
+
6
.
1 + (−q)n
θ3 (q 3 )
n=1
(5.0.27)
192
Lattice sums and Ramanujan’s modular equations
Now we apply alternate sign and Poisson transformations to obtain the following
set of eight third-order modular equations of dimension 2:
θ3 3
θ4 3
θ2 3 (q 3 )
θ5 3 (q 3 )
−
1
⇐
S
⇒
−
1
⇐
P
⇒
⇐
S
⇒
θ2
θ5
θ3 (q 3 )
θ4 (q 3 )
⇑
⇑
P
P
⇓
⇓
θ5 3
θ4 3 (q 3 )
θ2 3
θ3 3 (q 3 )
⇐S⇒
−1 ⇐S ⇒
−1 ⇐P ⇒
θ3
θ4
θ2 (q 3 )
θ5 (q 3 )
These results are displayed in Table A.2 together with their associated Mellin
transforms. A similar analysis was carried out on (5.0.23) and (5.0.24) and again
eight members of this set were found and are shown in Table A.3.
Another approach to third-order modular equations was made by Borwein
et al. [2]. Relations involving the expressons a(q), b(q), c(q), A = a(q 2 ), B =
b(q 2 ), C = c(q 2 ), where
2
2
a(q) =
q m +mn+n ,
(5.0.28)
2
2
b(q) =
ωn−m q m +mn+n ,
(5.0.29)
2
2
c(q) =
q (m+1/3) +(m+1/3)(n+1/3)+(n+1/3) ,
(5.0.30)
with ω = exp(2πi/3), were considered. A connection between the work in [2]
and that described here is briefly given. First, a(q) is given by its Lambert series as
∞ q 3n+1
q 3n+2
,
a(q) = 1 + 6
−
1 − q 3n+1
1 − q 3n+2
n=0
and it can be shown that
θ4 3 (q)
= 2a(q 2 ) − a(q)
θ4 (q 3 )
and
θ4 3 (q 3 )
= 13 a(q) + 23 a(q 2 ).
θ4 (q)
(5.0.31)
It is simple to show that these results agree with those given in Table A.2 by comparing their Mellin transforms. It is easily seen that M[a(q)−1] = 6L 1 (s)L −3 (s)
and M[a(q 2 ) − 1] = 2−s 6L 1 (s)L −3 (s). So, from (5.0.31) we have
M
θ4 3 (q)
− 1 = 21−s 6L 1 (s)L −3 (s) − 6L 1 (s)L −3 (s)
θ4 (q 3 )
= −6(1 − 21−s L 1 (s)L −3 (s),
M
θ4 3 (q 3 )
− 1 = 2L 1 (s)L −3 (s) + 4 × 2−s L 1 (s)L −3 (s)
θ4 (q)
= 2(1 + 21−s )L 1 (s)L −3 (s),
Lattice sums and Ramanujan’s modular equations
193
both of which agree with the results in Table A.2. For many other versions of third
order modular equations the original paper [2] should be consulted.
Two fifth-order equations of dimension 6 ((8.i) and (8.ii) in C.19) are given
below. These are
q
22 q 2
−
1 − q2
1 − q4
2
4
4 q
52 q 5
+
−
+ ··· ,
1 − q8
1 − q 10
q
2q 2
3q 3
qψ 3 (q)ψ(q 5 ) − 5q 2 ψ(q)ψ 3 (q 5 ) =
−
−
1 − q2
1 − q4
1 − q6
4q 4
6q 6
+
+
− ··· .
8
1−q
1 − q 12
qψ 5 (q)ψ(q 3 ) − 9q 2 ψ(q)ψ 5 (q 3 ) =
Treated as described above, a set of eight results was obtained and are given in
Table A.4. In Table A.5 some seventh-order results, from Entry 17 in C.19, are
given, and also relations for mixed modular equations of order 3, 5, and 15 derived
from Entry 10 in C.20. It is evident from these examples how groups of four and
eight members may be formed from any modular equation.
It is manifest that if we are able to show that
(n 2 + 3m 2 )−s = 2(1 + 21−2s )L 1 L −3 ,
(5.0.32)
in a way independent of Ramanujan’s methods then, by performing an inverse
Mellin transform on (5.0.31), we immediately retrieve the modular equation
θ3 θ3 (q ) − 1 = 2
3
∞
χ −3 q n
n=1
1 + (−q)n
.
Then, using the S and P operations, other modular relations are derivable. Now
the systematic evaluation of the sums
S = S(a, b, c) = S(a, b, c : s) =
(am 2 + bmn + cn 2 )−s ,
S1 = S1 (a, b, c) = S1 (a, b, c : s) =
(−1)m (am 2 + bmn + cn 2 )−s ,
(−1)n (am 2 + bmn + cn 2 )−s ,
S2 = S2 (a, b, c) = S2 (a, b, c : s) =
S1,2 = S1,2 (a, b, c) = S1,2 (a, b, c : s) =
(−1)m+n (am 2 + bmn + cn 2 )−s ,
where a, b, and c are integers, has been referred to in Section 1.4. It was conjectured there that if the binary quadratic form am 2 + bmn + cn 2 = (a, b, c) is such
that its reduced forms are disjoint, i.e., there is just one reduced form per genus,
then S is expressible as a linear sum of pairs of products of Dirichlet L-series.
A prescription for finding these pairs of Dirichlet series was given by Zucker
and Robertson [5]. Though counter-examples to this conjecture have been found,
194
Lattice sums and Ramanujan’s modular equations
they appear to be flukes. See Zucker and Robertson [6] for details. The principal
solutions of all such cases of one reduced form per genus are given in Section 1.4.
Thus the result
(n 2 + 7m 2 )−s = 2(1 − 21−s + 21−2s )L 1 L −7
is given there. By Mellin inversion this immediately gives
θ3 θ3 (q ) − 1 = 2
7
∞
χ −14b q n
n=1
1 + (−q)n
,
and then using S and P operations the other entries for seventh-order modular
equations may be found, as shown in Table A.5. Again, using Mellin transforms
it is simple to show that
θ3 θ3 (q 7 ) + θ4 θ4 (q 7 ) − θ2 θ2 (q 7 ) = 2θ4 (q 2 )θ4 (q 14 ) = 2 θ3 θ4 θ3 (q 7 )θ4 (q 7 ),
or
2
θ3 θ3 (q 7 ) + θ4 θ4 (q 7 ) = θ2 θ2 (q 7 ),
which is a well-known form of the seventh-order modular equation.
Some quite high-order modular equations may also be deduced. Thus it was
shown in [5] that if λ = (1+λ)/4 then for λ = 2, 3, 5, 11, 17, 41, corresponding
to λ = 7, 11, 19, 43, 67, 163,
S(1, 1, λ ) = 2L 1 L −λ .
It was also shown that
2S(1, 1, λ ) = 22s [S(1, 0, λ) + S1,2 (1, 0, λ)].
Hence
22s [S(1, 0, λ) + S1,2 (1, 0, λ)] = 4L 1 L −λ .
The inverse Mellin transform of the latter yields
θ3 (q 1/4 )θ3 (q λ/4 ) + θ4 (q 1/4 )θ4 (q λ/4 ) − 2 = 2 θ3 θ3 (q λ ) + θ2 θ2 (q λ ) − 1
=2
∞
χ −λ q n
n=1
1 − qn
.
Mixed modular relations may also be similarly obtained. Thus from [5] we have
(m 2 + 15n 2 )−s = (1 − 21−s + 21−2s )L 1 L −15 + (1 + 21−s + 21−2s )L −3 L 5 ,
whilst from [6] we obtain
(3m 2 + 5n 2 )−s = (1 − 21−s + 21−2s )L 1 L −15 − (1 + 21−s + 21−2s )L −3 L 5 .
Lattice sums and Ramanujan’s modular equations
195
Adding these together yields
(m 2 +15n 2 )−s +
(3m 2 +5n 2 )−s = 2(1−21−s +21−2s )L 1 L −15 . (5.0.33)
Then, taking the inverse Mellin transform of (5.0.33) immediately gives the mixed
modular equation of Entry (10.vi) in C.20. Once more the S and P operations
yield the other results given in Table A.5. Entries (9.i–iv) in C.20 may then be
simply deduced. Another example from [6] is
S(1, 0, 14) + S(2, 0, 7) = L 1 L −56 + L −7 L 8 ,
which yields mixed modular equations of orders 2, 7, and 14 in the form
θ3 θ3 (q 14 ) + θ3 (q 2 )θ3 (q 7 ) − 2 =
∞
χ −56 q n
n=1
1 − qn
+
χ −7 (q n q 3n )
1 + q 4n
.
Once again the use of S and P transformations would yield further results. It is
clear from these examples that, given solutions to (5.0.32), many modular and
mixed modular equations in terms of Lambert series may be generated.
Since the Mellin transforms of modular equations give relations between multidimensional sums and Dirichlet L-series, such Mellin transforms may often be
evaluated in terms of simple transcendental constants – a procedure of which
Ramanujan, with his love of numbers, would surely have approved. For amusement we give some examples. A simple case is that of the fourth result of
Table A.1, which may be translated as
4
∞
∞ −∞ −∞
(−1)m+n
s = 4L −4 (s)L 12 (s).
(2m − 12 )2 + 3(2n − 12 )2
√ √
For s = 1 the value of the right-hand side of the above is just π log(2 + 3)/ 3.
Similarly, using the seventh result in Table A.1 we have, for the four-dimensional
sum,
(−1)m+n+ p+r
= −4(1 − 22−s )(1 − 31−s )L 1 (s)L 1 (s − 1).
(m 2 + n 2 + 3 p 2 + 3r 2 )s
(5.0.34)
Putting s = 1 and s = 2 one has
(−1)m+n+ p+r
= −2 log 3,
m 2 + n 2 + 3 p 2 + 3r 2
(−1)m+n+ p+r
4
= − π 2 log 2.
2
2
2
2
2
9
(m + n + 3 p + 3r )
By differentiating both sides of (5.0.34) with respect to s, expressions such as
(−1)m+n+ p+r log(m 2 + n 2 + 3 p 2 + 3r 2 )
π
= (2 log 3) log √ − γ ,
m 2 + n 2 + 3 p 2 + 3r 2
2 3
where γ is Euler’s constant, may be obtained.
196
Lattice sums and Ramanujan’s modular equations
A more complicated example is as follows. Let
(−1)m+n+ p+r
and
+ n 2 + p 2 + 5r 2 )s
(−1)m+n+ p+r
B(s) =
.
2
(m + 5n 2 + 5 p 2 + 5r 2 )s
A(s) =
(m 2
Now, it is clear that
A(s) = M θ4 3 θ4 (q 5 ) − 1
and
B(s) = M θ4 θ4 3 (q 5 ) − 1 ,
and, from Table A.4 we have
A(s) − B(s) = −4(1 − 22−s )L 1 (s − 1)L 5 (s),
5B(s) − A(s) = −4(1 + 22−s )L 1 (s)L 5 (s − 1).
Hence
A(s) = −5(1 − 22−s )L 1 (s − 1)L 5 (s) − (1 + 22−s )L 1 (s)L 5 (s − 1),
B(s) = −(1 − 22−s )L 1 (s − 1)L 5 (s) − (1 + 22−s )L 1 (s)L 5 (s − 1).
For s = 1 and 2, A(s) and B(s) take particularly simple forms; we have the
following results:
1
L 1 (0) = − ,
2
√ 1+ 5
2
L 5 (1) = √ log
,
2
5
L 5 (0) = 0,
√ 1
+
5
L 5 (0) = log
,
2
π2
,
6
4π 2
L 5 (2) = √ ,
25 5
L 1 (2) =
obtained using the standard limiting procedures
√ 1+ 5
.
lim L 1 (s)L 5 (s − 1) = log
s→1
2
lim (1 − 2
1−s
s→1
)L 1 (s) = log 2,
Consequently, we have
√ 1+ 5
A(1) = −(3 + 5) log
,
2
√
1
1+ 5
B(1) = − 3 + √ log
,
2
5
√ 1+ 5
−4π 2 log 2 1
+ log
,
A(2) = √
5
6
2
5
√
−4π 2 log 2 1
1+ 5
.
B(2) = √
+ log
25
6
2
5
√
Clearly many more such results may be established.
5.1 Commentary: The modular machine
197
5.1 Commentary: The modular machine
As described in [2] there is a modular machine which, given sufficient knowledge of the ambient modular function theory, allows one to prove experimentally
discovered equations of the kind described in this chapter. Various extensions are
given in [3].
We paraphrase the modular approach as follows. It is possible to both find and
prove modular identities entirely mechanically. We will illustrate how this works
by proving the identity
a 3 (q) = b3 (q) + c3 (q),
(5.1.1)
where, for ω = e2πi/3 ,
a(q) :=
b(q) :=
c(q) :=
∞
qn
n,m=−∞
∞
n,m=−∞
∞
2 +nm+m 2
ωn−m q n
q (n+1/3)
,
2 +nm+m 2
,
2 +(n+1/3)(m+1/3)+(m+1/3)2
.
n,m=−∞
From the Lambert series
∞ q 3n+1
q 3n+2
,
a(q) = 1 + 6
−
1 − q 3n+1
1 − q 3n+2
(5.1.2)
n=0
we see that a(q) (in the variable τ , where q = e2πiτ ) is an Eisenstein series
of weight 1 and character χ (d) = (d | 3) (the Legendre symbol modulo 3)
for the congruence subgroup 0 (3). It is well known that if f (τ ) is a modular form on 0 (N ) then f (Mτ ) is a modular form on 0 (N M). From the
equations
1
3
a(q 3 ) − a(q),
2
2
1
1
c(q) = a(q 1/3 ) − a(q),
2
2
b(q) =
(5.1.3)
(5.1.4)
it follows that a(q 3 ), b(q 3 ), c(q 3 ), a(q 6 ), b(q 6 ), c(q 6 ) are all entire modular
forms of weight 1 (and trivial character) on some congruence subgroup G, where
(3) ∩ 0 (54) ⊂ G ⊂ 198
Lattice sums and Ramanujan’s modular equations
(we are working in the variable q 3 to give c a Taylor series expansion at i∞).
Here, as usual,
α β :=
αδ
−
βγ
=
1,
α,
β,
γ
,
δ
∈
Z
,
γ δ α β
(N ) :=
∈ α ≡ δ ≡ 1 and β ≡ γ ≡ 0 mod N ,
γ δ
α β
0 (N ) :=
∈ γ ≡ 0 mod N .
γ δ
The indices satisfy
[ : (N )] = N 3
p|N
and
[ : 0 (N )] = N
1−
1 p2
1
.
1+
p
p|N
It follows that
[ : G] ≤ [ : (3)] × [ : 0 (54)] ≤ 24 × 108.
The standard theory can be found in [4].
Now suppose that
P := P(a, b, c, A, B, C)
(5.1.5)
is a homogeneous polynomial of degree N in the six variables a := a(q), b :=
b(q), c := c(q), A := a(q 2 ), B := b(q 2 ), C := c(q 2 ). Then P(q) is an entire
modular form of weight N on G and hence can have exactly N [ : G]/12 zeros
in a fundamental region (counted in the local variables at the cusps). In particular
P can have a zero of order at most 216N at τ = i∞. In other words, if the
q-expansion of P vanishes through the first 216N + 1 terms then P ≡ 0.
It is now a straightforward matter to generate a basis for all homogeneous
identities of the type (5.1.5) for a fixed N . One expands the six functions
a, b, c, A, B, C as q-series to some fixed order that is greater than the number
of monomials in the expansion of (x1 + x2 + x3 + x4 + x5 + x6 ) N . One then solves
the linear problem of finding a basis of identities to this fixed order. This must
now be a superset of the desired identities. One then verifies that the q-expansion
of each basis element vanishes through 216N + 1 terms, which proves that the
alleged identity is a true identity (and not just an identity to a fixed number of
terms). Since this is all done in exact integer arithmetic in a symbolic manipulation
package this constitutes both a derivation and a proof.
We will illustrate this for N = 3. What follows are the basis elements for all
homogeneous cubic relations in a, b, c, A, B, C.
5.2 Commentary: A cubic theta function identity
(1) A(c2 − aC − 2AC)
(2) B(c2 − aC − 2AC)
(3) c(bB − Aa + cC)
(4) B(bB − Aa + cC)
(5) C(bB − Aa + cC)
(6) A(c A − ac + 2C 2 )
(7) B(c A − ac + 2C 2 )
(8) C(c A − ac + 2C 2 )
(9) −b(c A − ac + 2C 2 )
(10) B(b A − 2B 2 + ab)
(11) C(b A − 2B 2 + ab)
(12) −A(−a B − b2 + 2AB)
(13) −B(−a B − b2 + 2AB)
(14) −C(−a B − b2 + 2AB)
(15) − B(Aa − 2bB − a 2 + 2A2 )
(16) − C(Aa − 2bB − a 2 + 2A2 )
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
199
−C 3 − B 3 + A3
4C 3 − 3acC + c3
2B 3 − 3b AB + b3
− bc A − bC 2 + cB 2
bcC − a B 2 + b A2
bc2 − b AC − 2B 2 C
b2 c − acB + 4BC 2
ab2 − a AB − 2bB 2 + 2A2 B
bcC + a 2 b − 3a B 2 + 2AB 2
2bBC − 3a AC + ac2 − 2A2 C
2bcB − 3ac A + a 2 c + 4AC 2
ab A − bcC + a B 2 − 2AB 2
bcB − ac A + aC 2 + 2AC 2
b AB − acC − 2B 3 + a 2 A
−b AB − acC + 2C 3 + a A2
−3b AB − 3acC + 4C 3 + 2B 3 + a 3
Thus we find 32 basis elements, presented in factored form. Since there is a
basis of six quadratic relations, many of them do factor. The verification of the
identities requires computing Taylor series of length 650. The entire calculation
took just a few minutes in Maple (in the early 1990s). Note that basis element
(17) is our cubic modular equation (5.1.1) while basis element (3) is a quadratic
identity given by Ramanujan. More careful analysis reveals the need to check only
a lower-degree estimate in the q-expansion; however, since the computations are
easy we have opted for the most straightforward estimates. Similar remarks apply
for other forms and other N . Note that (1) and (12) combine to give
1 b2
1 c2
−
,
2C
2 B
while from (11) and (12) we may solve for A, B from a, b, and c.
To summarize, the symbolic manipulation of series and products for theta
functions – informed by the modular machine – is an enormously effective way of
formalizing knowledge gleaned from numerical experimental tools such as PSLQ.
a=
5.2 Commentary: A cubic theta function identity
Let us return to
a(q) =
qm
2 +nm+n 2
,
m,n
and
c(q) =
b(q) =
2
2
(ω)n−m q m +nm+n ,
m,n
m,n
q (m+1/3)
2 +(n+1/3)(m+1/3)+(n+1/3)2
,
200
Lattice sums and Ramanujan’s modular equations
where ω = e2iπ/3 ; these expressed were given in (5.0.28). Then one may convince
oneself computationally that for |q| < 1 the functions a, b, and c solve Fermat’s
equation of degree 3,
a 3 = b3 + c3 .
(5.2.1)
The neatest proof of (5.2.1) [2] relies on expressing b and c as q-products. For
example, b(q) starts
3 2 3 3 2
1 − q3
1 − q4
1 − q5
1 − q6
(1 − q)3 1 − q 2
3 2
&3 %
1 − q8
1 − q9 ,
× 1 − q7
which tells us the closed form, whereas
&21 %
&345 %
&8906 %
&250257
%
1 − q4
1 − q6
1 − q8
1 − q2
a(q) =
%
&76 %
&1734 %
&46662 %
&1365388
1 − q5
1 − q7
1 − q9
(1 − q)6 1 − q 3
clearly has no such nice product factorization.
As sums, a and b look quite similar:
a(q) = 1 + 6q + 6q 3 + 6q 4 + 12q 7 + 6q 9 + 6q 12 + 12q 13 + 6q 16
+ 12q 19 + O q 21
and
b(q) = 1 − 3q + 6q 3 − 3q 4 − 6q 7 + 6q 9 + 6q 12 − 6q 13 − 3q 16
− 6q 19 + O q 21 .
It is instructive to write code to replicate these results and to explore c, which, like
b, has both a nice q-product and a Lambert series.
Note that a 3 = b3 + c3 is actually an identity for six-dimensional sums.
Moreover, we have the following hypergeometric parameterization [2]:
2 F1
1 2
3, 3
1
; ab3 (q)
(q)
3
= a(q),
which implies a cubic transformation for the hypergeometric function that was
known to Ramanujan.
References
[1] B. C. Berndt. Ramanujan’s Notebooks Part III. Springer-Verlag, New York, 1991.
[2] J. M. Borwein, P. B. Borwein, and F. G. Garvan. Some cubic modular identities of
Ramanujan. Trans. Amer. Math. Soc., 343:35–47, 1994.
[3] M. Hirschhorn, F. Garvan, and J. M. Borwein. Cubic analogues of the Jacobian theta
function (q, z). J. Math. Phys., 19:1064, 1978.
References
201
[4] B. Schoeneberg. Elliptic Modular Functions. Springer-Verlag, New York, 1974.
[5] I. J. Zucker and M. M. Robertson. Systematic approach to the evaluation of
2
2 −s
(m,n=0,0) (am + bmn + cn ) . J. Phys. A, 9:1215–1225, 1976.
[6] I. J. Zucker and M. M. Robertson. Further aspects of the evaluation of
2
2 −s
(m,n=0,0) (am + bmn + cn ) . Math. Proc. Camb. Phil. Soc., 95:5–13, 1984.
6
Closed-form evaluations of threeand four-dimensional sums
In a 1980 survey of lattice sums [10], the following remarks may be found:
‘whereas in two dimensions it appears we know precisely when we can express
a lattice sum exactly, and also in four and higher even dimensions where many
exact results are obtainable, in contrast, in three dimensions there is a paucity of
exact results. Indeed, only two are known . . . ’. The situation has changed somewhat since then and many more three-dimensional lattice sums have now been
evaluated in closed form. These new results have been obtained by generalizing
the Jacobian theta series and products described previously in Section 1.3. An
account of these extensions is now given, and their properties described. Connections between series and products are discussed, and a notation developed to
express them in a transparent way which enables them to be simply manipulated.
Later in this chapter some recent developments on four-dimensional sums are also
delineated.
6.1 Three-dimensional sums
Four sets of series are considered (in this chapter, we often denote sums and prod
ucts simply by
and
respectively, when the summation or multiplication
variable(s) are clear from the context). They are as follows: the set ,
2
2
θ̄ (k, l) =
(−1)n q (kn+l) , (6.1.1)
θ (k, l) =
q (kn+l) ,
2
2
φ̄(k, l) =
(−1)n(n+1)/2 q (kn+l) ;
φ(k, l) =
(−1)n(n−1)/2 q (kn+l) ,
the set X ,
2π n
2π n
2
(kn+l)2
n
cos
q
q (kn+l) ,
χ (k, l) =
,
χ̄ (k, l) =
(−1) cos
3
3
2π n
2
n(n−1)/2
cos
q (kn+l) ,
ψ(k, l) =
(−1)
3
2π n
2
n(n+1)/2
ψ̄(k, l) =
(−1)
cos
q (kn+l) ;
(6.1.2)
3
6.1 Three-dimensional sums
the set ,
θ (k, l) =
203
2
2
(kn + l)q (kn+l) ,
θ̄ (k, l) =
(−1)n (kn + l)q (kn+l) ,
2
φ (k, l) =
(−1)n(n−1)/2 (kn + l)q (kn+l) ,
(6.1.3)
2
φ̄ (k, l) =
(−1)n(n+1)/2 (kn + l)q (kn+l) ;
the set X ,
2π n
2π n
2
(kn+l)2
n
(kn+l)q
q (kn+l) ,
χ (k, l) =
cos
, χ̄ (k, l) = (−1) cos
3
3
2π
n
2
(kn + l)q (kn+l) ,
(−1)n(n−1)/2 cos
(6.1.4)
ψ (k, l) =
3
2π n
2
ψ̄ (k, l) =
(−1)n(n+1)/2 cos
(kn + l)q (kn+l) .
3
In all these series denotes summation over all integers n from −∞ to ∞. All
the previous Jacobian theta series are accommodated by this notation; thus
θ3 = θ (1, 0) = θ (1, 1),
θ4 = θ̄ (1, 0),
θ2 = θ (2, 1)|q 1/4 ,
θ5 = 2θ̄ (4, 1)|q 1/4 .
Here the vertical bar followed by q t means that q t replaces q in the original
expression. These series have expansion properties, typically
θ (k, l) = θ (2k, l) + θ (2k, k + l) = θ (3k, l) + θ (3k, k + l) + θ (3k, 2k + l) + · · ·
θ̄ (k, l) = θ (2k, l) − θ (2k + k, l) = θ̄ (3k, l) − θ̄ (3k, k + l) + θ̄ (k, 2k + l) + · · ·
φ(k, l) = θ̄ (2k, l) + θ̄(2k, k + l),
φ(k, l) = θ̄ (2k, l) − θ̄(2k, k + l).
(6.1.5)
These expansions will allow us sometimes to convert new series into more
conventional notation. For example, expanding θ (1, 1) we have
θ (1, 1) = θ (3, 1) + θ (3, 2) + θ (3, 3).
But θ (3, 1) = θ (3, 2) and θ (3, 3) = θ (1, 1)|q 3 = θ3 (q 9 ), and we thus have
1
θ3 − θ3 (q 9 ) .
θ (3, 1) =
2
We further define two infinite products
Q(k, l) =
(1 − q kn−l ),
Q̄(k, l) =
(1 + q kn−l ),
(6.1.6)
where implies the product over all n from 1 to ∞. These also have expansion
properties; thus
Q(k, l) = Q(2k, l)Q(2k, 2k, k + l) = Q(3k, l)Q(3k, k + l)Q(3k, 2k + l) = · · ·
(6.1.7)
204
Closed-form evaluations of three- and four-dimensional sums
with a similar expression for Q̄(k, l). It may be shown that all members of may be expressed in terms of Q(k, l) and Q̄(k, l). For example, starting with the
fundamental Jacobi identity
2
(1 − q 2n )(1 + xq 2n−1 )(1 + x −1 q 2n−1 ),
xnqn =
2
2
replacing q by q k , x by q 2kl , and multiplying both sides by q l , one obtains
immediately
2
θ (k, l) = q l Q(2k, 0) Q̄(2k, k + 2l) Q̄(2k, k − 2l)|q k .
(6.1.8)
With similar but more complicated substitutions we may deduce that
2
θ̄ (k, l) = q l Q(2k, 0)Q(2k, k + 2l)Q(2k, k − 2l)|q k ,
2
φ(k, l) = q l Q(4k, 0) Q̄(4k, 2k)Q(4k, k − 2l) Q̄(4k, 3k − 2l)
Q̄(4k, k + 2l)Q(4k, 3k + 2l)|q k ,
2
φ̄(k, l) = q l Q(4k, 0) Q̄(4k, 2k) Q̄(4k, k − 2l)Q(4k, 3k − 2l)Q(4k, k + 2l)
Q̄(4k, 3k + 2l)|q k .
Some further notational additions are introduced here. The Euler partition func
tion Q(k, 0) = (1−q kn ) will be denoted by (k) and Jacobi’s favourite products,
Q0 =
Q3 =
(1 − q 2n ),
Q1 =
(1 + q 2n ),
Q2 =
(1 + q 2n−1 ),
(6.1.9)
(1 − q 2n−1 ),
will be denoted by w, x, z, y respectively. A dictionary is easily established
between the two sets of notations; thus
w = (2),
x=
(4)
,
(2)
y=
(1)
,
(2)
z=
(2)2
,
(1)(4)
(6.1.10)
from which the well-known result x yz = 1 is immediately obtained. Expressions
for and X in terms of (k) will be termed Eulerian, whereas expressions in terms
of w, x, y, z will be called Jacobian. It is also convenient to introduce here a slight
modification of the Euler partition function, namely the Dedekind eta function,
which is defined as
η(k) = q k/24 (1 − q kn ) = q k/24 (k) := [k].
Three operations which will be applied to sums and products introduced here
will now be described. The first is the operation S (the sign transform), which
changes q to −q in any expression. We have the following results:
6.1 Three-dimensional sums
Sw = w,
S x = x,
S y = z,
S(k) = (k) for k an even integer;
S(k) =
205
Sz = y;
(2k)3
(k)(4k)
(6.1.11)
for k an odd integer.
(See above for the meaning of (k).) For example, these results lead to the
following:
Sθ3 = θ4 ,
S
θ2 (q 1/2 )
θ5 (q 1/2 )
=
.
q 1/8
q 1/8
The second operation, P, is a Poisson transformation (cf. the Poisson summation formula). This transformation changes expressions in term of q = e−π t into
those in terms of a complementary variable, ρ = e−π/t . The fundamental result
for a q-series is
(
2π nl
1
2 2
cos
ρ n /k .
Pkθ (k, l) =
(6.1.12)
t
k
The other thetas transform as below:
(
πl
1
2
2
Pk θ̄ (k, l) =
cos (2n + 1)
ρ (2n+1) /4k ,
(6.1.13)
t
k
(
π
k + 2l
1 2
2
cos(2n + 1)
cos(2n + 1)π
ρ (2n+1) /16k ,
Pkφ(k, l) =
t
4
4k
(6.1.14)
(
π
k + 2l
1
2
2
Pk φ̄(k, l) =
sin (2n + 1)
sin(2n + 1)π
ρ (2n+1) /16k .
t
4
4k
(6.1.15)
For transforming products the most important result is expressed in terms of the
Dedekind eta function. This transforms as follows:
(
2
1
Pη[2k : q] =
η
:ρ ,
(6.1.16)
kt
k
which enables one to transform any result in Eulerian form to another in that
form, since the P operation is multiplicatively transitive. That is, P[a][b][c] · · ·
= P[a]P[b]P[c] · · · .
The Jacobian quantities in (6.1.10) transform thus:
P q 1/12 x = 2−1/2 ρ −1/24 y,
P q 1/12 w = ρ 1/12 w,
(6.1.17)
P q −1/24 z = ρ 1/24 z.
P q −1/24 w = 21/2 ρ 1/12 x,
Expressions related by P-transformations will be termed Poisson conjugates.
Since q and ρ are completely interchangeable, a Poisson transform of a q-relation
will yield another q-relation unless the expression is self-conjugate.
206
Closed-form evaluations of three- and four-dimensional sums
The third operation which will be applied is the Mellin transform, M,
defined by
∞
1
t s−1 f (t) dt.
M[ f (t)] =
(s) 0
This operation can be applied to any q-series after replacing q by e−t . As
examples of such a transformation, consider first
2
2
2
2
θ̄ (6, 1) =
(−1)n q (6n+1) = q − q 5 − q 7 + q 11 + · · ·
Replacing q by e−t and taking the Mellin transform, we have
∞
1 2
(−1)n
M θ̄(6, 1) =
t s−1 e−t (6n+1) dt
(s)
0
(−1)n
=
= 1 − 5−2s − 7−2s + 11−2s + · · · = L 12 (2s),
(6n + 1)2
where L 12 is a positive-parity Dirichlet L-series of argument 2s. Similarly, if we
perform the same operations on θ̄ (6, l), we have
2
2
2
2
θ̄ (6, 1) =
(−1)n (6n + 1)q (6n+1) = q + 5q 5 − 7q 7 − 11q 11 + · · · ,
M θ̄ (6, 1) = 1 + 52s−1 − 72s−1 − 112s−1 + · · · = (1 + 31−2s )L −4 (2s − 1),
where L −4 is a negative-parity L-series of argument 2s − 1. In general, Mellin
transforms of members of and X always yield positive-parity L-series of degree
2s, whereas Mellin transforms of members of and X give negative-parity
L-series of degree 2s − 1.
The reason why this preliminary discussion was presented was to explain how
members of , X , , and X may be expressed in a particularly apt form of
infinite product. This will be illustrated with the example of θ (3, 1). From (6.1.8),
θ (3, 1) = q Q(6, 0) Q̄(6, 5) Q̄(6, 1)|q 3 .
Now
Q̄(1, 0) = Q̄(6, 0) Q̄(6, 1) Q̄(6, 2) Q̄(6, 3) Q̄(6, 4) Q̄(6, 5),
so
Q̄(6, 5) Q̄(6, 1) =
Q̄(1, 0)
.
Q̄(6, 0) Q̄(6, 2) Q̄(6, 3) Q̄(6, 4)
But
Q̄(6, 0) Q̄(6, 2) Q̄(6, 4) = Q̄(3, 0) Q̄(3, 1) Q̄(3, 2)|q 2 = Q̄(1, 0)|q 2 =
and
Q̄(1, 0) =
(2)
,
(1)
Q̄(6, 3) = Q̄(2, 1)|q 3 =
(6)2
.
(3)(12)
(4)
(2)
6.1 Three-dimensional sums
207
Combining these together leads to
θ (3, 1) = q
(6)2 (9)(36)
(3)(12)(18)
or
θ (3, 1)|q 1/3 = q 1/3
(2)2 (3)(12)
. (6.1.18)
(1)(4)(6)
Now perform a sign transform on both side of (6.1.18); thus
S
θ̄ (3, 1)|q 1/3
(2)2 (3)(12)
θ (3, 1)|q 1/3
=
=
S
(1)(4)(6)
q 1/3
q 1/3
and, following the prescription given in (6.1), one obtains
θ̄ (3, 1)|q 1/3 = q 1/3
(1)(6)2
.
(2)(3)
Similarly, apply the Poisson transform to θ (3, 1). From (6.1.12),
1 1
2π n n 2 /9
Pθ (3, 1) = √
= √ χ (1, 0)|ρ 1/9 .
cos
ρ
3
3 t
3 t
From (6.1.18) we have
' ' 2 1
2
1
2
4
1
3
9
9
18
3
9
9
[6]2 [9][36]
1 3
Pθ (3, 1) = P
= √ ' ' ' .
[3][12][18]
t
2
1
1
4
1
2
3
6
9
3
Equating these two expressions for Pθ (3, 1) and replacing ρ 1/9 by q, we obtain
χ (1, 0) =
(1)(4)(6)2
.
(2)(3)(12)
This process may be continued by taking a Poisson transform of θ̄(3, 1) and a
sign transform of χ (1, 0) and then alternating these two transforms, forming the
following closed set:
θ(3, 1)|q 1/3
⇑
⇐S⇒
θ̄ (3, 1)|q 1/3
⇐P⇒
2χ(2, 1)|q 1/8
⇐S⇒
P
⇓
χ(1, 0)
2ψ̄(2, 1)|q 1/8
⇑
P
⇐S⇒
χ̄(1, 0)
⇐P⇒
θ(6, 1)|q 1/24
⇐S⇒
⇓
φ(6, 1)|q 1/24
Following the procedures just described, all the members of sets and X given
in Table 6.1 in Eulerian and Jacobian forms were constructed. In Table 6.2 the
sign, Poisson, and Mellin transforms of these series are also given. As previously
described, the Mellin transforms of all the series in Table 6.1 yield a Dirichlet
L-series whose character depends on the series. Because L-series of different
character are algebraically independent, the independence or otherwise of the
various elements of and X may easily be established.
Since the history of expressing infinite series in product form and vice versa
goes back to Euler, it has not been possible to determine with certainty the original
208
Closed-form evaluations of three- and four-dimensional sums
Table 6.1 Members of and of X which may be expressed in Eulerian
and Jacobian forms
Series
Conventional
Eulerian
Jacobian
Equation
θ (1, 0)
θ3
wz 2
T(1.1)
θ̄(1, 0)
θ4
wy 2
T(1.2a)
θ̄(1, 0)|q 2
θ4 (q 2 )
wyz
T(1.2b)
θ (2, 1)|q 1/8 =
2θ (4, 1)|q 1/8
θ2 (q 1/2 )
(2)5
(1)2 (4)2
(1)2
(2)
(2)2
(4)
2
2q 1/8 (2)
(1)
2q 1/8 wx z
T(1.3a)
θ (2, 1)|q 1/4 =
2θ (4, 1)|q 1/4
θ̄(4, 1)|q 1/8
θ2
2q 1/4 (4)
(2)
2q 1/4 wx 2
T(1.3b)
1
1/2 )
2 θ5 (q
1
1/3 ) − θ (q 3 )
3
2 θ3 (q
1
3
1/3 )
2 θ4 (q ) − θ4 (q
1
6
2/3 )
2 θ4 (q ) − θ4 (q
1
1/6 ) − θ (q 3/2 )
2
2 θ2 (q
q 1/8 (1)(4)
(2)
q 1/8 wx y
T(1.4)
q 1/3 z|wx y|q 3
T(1.5)
q 1/3 y|wx z|q 3
T(1.6a)
q 2/3 yz|wx 2 |q 3
T(1.6b)
q 1/24 x z|wy 2 |q 3
T(1.7a)
q 1/12 x|wyz|q 3
T(1.7b)
θ (3, 1)|q 1/3
2
q 1/3
(2)2 (3)(12)
(1)(4)(6)
q 1/24 x y|wz 2 |q 3
T(1.8)
θ̄(6, 1)|q 1/24
2
q 1/3 (1)(6)
(2)(3)
2
q 1/3 (2)(12)
(4)(6)
2
q 1/24 (2)(3)
(1)(6)
2
q 1/12 (4)(6)
(2)(12)
5
q 1/24 (1)(4)(6)
(2)2 (3)2 (12)2
q 1/24 (1)
q 1/24 wy
T(1.9a)
θ̄(6, 1)|q 1/12
q 1/12 (2)
q 1/12 w
T(1.9b)
θ̄(6, 1)|q 1/6
q 1/6 (4)
q 1/6 wx
T(1.9c)
φ̄(6, 1)|q 1/24
(2)3
q 1/24 (1)(4)
(1)(4)(6)2
(2)(3)(12)
(2)2 (3)
(1)(6)
(4)2 (6)
(2)(12)
2 (6)
q 1/8 (1)
(2)(3)
2 (12)
q 1/4 (2)
(4)(6)
2
q 1/8 (2)2 (3)(12)
(1) (4)2 (6)2
q 1/24 wz
T(1.10)
wx y|z|q 3
T(1.11)
wx z|y|q 3
T(1.12a)
wx 2 |yz|q 3
T(1.12b)
q 1/8 wy 2 |x z|q 3
T(1.13a)
q 1/4 wyz|x|q 3
T(1.13b)
q 1/8 wz 2 |x y|q 3
T(1.14)
θ̄(3, 1)|q 1/3
θ̄(3, 1)|q 2/3
θ (6, 1)|q 1/24
θ (6, 1)|q 1/12
φ(6, 1)|q 1/24
χ (1, 0)
χ̄(1, 0)
χ̄(1, 0)|q 2
2χ (2, 1)|q 1/8
2χ (2, 1)|q 1/4
2ψ̄(2, 1)|q 1/8
1
1/3 ) − θ (q 3 )
2
2 θ2 (q
1 θ (q 1/3 ) + θ (q 3 )
5
2 5
1 3θ (q 9 ) − θ
3
3
2
1 3θ (q 9 ) − θ
4
4
2
1
18
2
2 3θ4 (q ) − θ4 (q )
1
1/2 ) − 3θ (q 9/2 )
2
2 θ2 (q
1
9
2 θ2 − 3θ2 (q )
1
1/2 ) + 3θ (q 9/2 )
5
2 θ5 (q
sources of the results given in Table 6.1. However, after a considerable literature
search it is believed that the following attributions of the results in Table 6.1 are
correct. Result (T1.l) is due to Jacobi; (T1.2) and (T1.3) are due to Gauss; (T1.9)
is Euler’s classic result, the very first of its kind; (T1.6) and (T1.7) were given
by Kac [14], who claims them as new and also gives (T1.11) and (T1.13) as new,
but those results go back to Ramanujan [18]. Results (T1.4) and (T1.10) as given
here are new, though (T1.4) is implied by [24]; however, [22] quotes some results
of Jacobi from which (T1.4) and (T1.l0) may be deduced. As far as we know
6.1 Three-dimensional sums
209
Table 6.2 The sign, Poisson, and Mellin transforms of some members of and X ; the argument of the L-series is (2s)
Series
Sign transform Poisson transform
Mellin transform
Equation
θ (1, 0) − 1
θ̄(1, 0) − 1
θ (1, 0) − 1
2L 1
T(2.1)
θ̄(1, 0) − 1
θ (1, 0) − 1
θ (2, 1)|q 1/4
θ (2, 1)|q 1/8
2θ̄(4, 1)|q 1/8
√
2θ̄(1, 0)|q 2
−2(1 − 21−2s )L
θ (3, 1)|q 1/3
θ̄(3, 1)|q 1/3
θ̄(3, 1)|q 1/3
T(2.2)
1
21+3s (1 − 2−2s )L 1
T(2.3)
3−1/2 χ (1, 0)|q 1/3
3s (1 − 3−2s )L
T(2.4)
θ (3, 1)|q 1/3
2 × 3−1/2 χ (2, 1)|q 1/12
3s (1 − 21−2s )(1 − 3−2s )L 1
T(2.5)
θ̄(4, 1)|q 1/8
θ (4, 1)|q 1/8
θ̄(4, 1)|q 1/8
T(2.6)
θ (6, 1)|q 1/2
φ(6, 1)|q 1/24
θ̄(6, 1)|q 1/24
φ̄(6, 1)|q 1/24
√ −1
6.3 χ̄ (1, 0)|q 2/3
√
2θ̄(6, 1)|q 1/6
8s L
φ(6, 1)|q 1/24
θ (6, 1)|q 1/24
3−1/2 ψ̄(2, 1)|q 1/24
24s (1 + 3−2s )L
φ̄(6, 1)|q 1/24
θ̄(6, 1)|q 1/24
φ̄(6, 1)|q 1/24
24s L 24
χ (1, 0)
χ̄(1, 0)
θ (3, 1)|q 1/9
−(1 − 31−2s )L
χ̄(1, 0)
χ (1, 0)
θ (6, 1)|q 1/36
(1 − 21−2s )(1 − 31−2s )L 1
2χ (2, 1)|q 1/8
2ψ̄(2, 1)|q 1/8
θ̄(3, 1)|q 2/9
8s (1 − 2−2s )(1 − 31−2s )L
2ψ̄(2, 1)|q 1/8
2χ (2, 1)|q 1/8
φ(6, 1)|q 1/72
8s (1 + 31−2s )L 8
1
8
24s (1 − 2−2s )(1 − 3−2s )L 1
T(2.7)
24s L 12
T(2.8)
T(2.9)
8
T(2.10)
T(2.11)
1
T(2.12)
1
T(2.13)
T(2.14)
(T1.5), (T1.8), (T1.12), and (T1.14) are new as given here, though [2] establishes
something equivalent to (T1.12) in terms of conventional notation.
In Table 6.3 we display results for and X in Eulerian and Jacobian form.
It has not been found possible to obtain these results by the systematic procedure
used for and X . Ad hoc methods were needed to derive the results given in
Table 6.3. One result, (T3.3), may be derived from a classical identity of Jacobi,
which is given in [12] as
∞
(−1)n (2n + 1)q n(n+1)/2 =
(1 − q n )3 = (1)3 .
(6.1.19)
n=0
A more symmetric form of (6.1.19) may be found by using the fact that, for any
function f,
∞
(−1)n (2n + 1) f (2n + 1) =
(4n + 1) f (4n + 1).
n=0
Thus, multiplying both sides of (6.1.19) by q 1/8 the latter may be written
2
(4n + 1)q (4n+1) /8 = q 1/8 (1)3 .
A method of finding in terms of Lambert series and recognizing these as
expressible as infinite products can produce desired results. This will be illustrated
¯ (k, 1). The Jacobi triple product identity may be written
now for 210
Closed-form evaluations of three- and four-dimensional sums
2
(−1)n a n q n = (2) (1 − aq 2n−1 )(1 − a −1 q 2n−1 ).
(6.1.20)
The following operations are carried out in this order: (i) replace a by a k ; (ii) multiply both sides of (6.1.20) by a l ; (iii) differentiate logarithmically with respect to
2
a, then multiply by a; (iv) replace q by q k and a by q 2l . Then it is found that
2
2
2
2
θ̄ (k, l)
q 2nk −k −2kl
q 2nk −k +2kl
=1+k
−
.
(6.1.21)
2
2
2
2
θ̄ (k, l)
1 − q 2nk −k −2kl
1 − q 2nk −k +2kl
Series such as those found on the right-hand side of (6.1.21) are known as Lambert series. In order to identify them, Mellin transforms are formed. For example,
consider k = 4, and 1 = 1. We have
∞
q 4n−3
q 4n−1
θ̄ (4, 1) 1/8
−
.
(6.1.22)
|q − 1 = 4
1 − q 4n−3
1 − q 4n−1
θ̄ (4, 1)
1
The Mellin transform of the right-hand side of (6.1.22) is 4L 1 (s)L −4 (s). But it is
a well-known result [12], going back to [16] and implicit in Jacobi, that
M[θ32 − 1] = M[θ 2 (1, 0) − 1] = 4L 1 (s)L −4 (s).
So, the right-hand side of (6.1.22) is just θ 2 (1, 0) − 1, and thus
θ̄ (4, 1)|q 1/8 = θ̄(4, 1)|q 1/8 θ 2 (1, 0) = q 1/8
(2)9
,
(1)3 (4)3
which after using the results (T1.1) and (T1.4) gives (T3.4).
Table 6.3 Members of and X expressed in Eulerian and Jacobian forms;
the argument of the L-series is (2s − 1)
Series
Eulerian
Jacobian
Mellin transform
Equation
θ (3, 1)|q 1/3
q 1/3 w 3 x 2 y 2
3s L −3
T(3.1)
q 1/3 w 3 x 2 z 2
3s (1 + 22−2s )L −3
T(3.2)
θ (4, 1)|q 1/8
2 (4)2
q 1/3 (1)(2)
5
q 1/3 (2)2
(1)
q 1/8 (1)3
q 1/8 w 3 y 3
23s L −4
T(3.3)
θ̄ (4, 1)|q 1/8
q 1/8
(2)9
(1)3 (4)3
5
q 1/24 (1)2
(2)
13
q 1/24 (2)
(1)5 (4)5
q 1/8 (1)3 + 3q 9/8 (9)Â č
q 1/8 w 3 z 3
23s L
T(3.4)
q 1/24 w 3 y 5
24s (1 + 21−2s )L −3
T(3.5)
q 1/24 w 3 z 5
24s L −24
T(3.6)
θ̄ (3, 1)|q 1/3
θ (6, 1)|q 1/24
φ (6, 1)|q 1/24
θ̄ (6, 1)|q 1/8
2χ̄ (2, 1)|q 1/8
9
9
− 3q 9/8 (18)
q 1/8 (2)
(1)3 (4)3
(9)3 (36)3
q 1/8 (1)3 + 9q 9/8 (9)3
2ψ (2, 1)|q 1/8
q 1/8
φ̄ (6, 1)|q 1/8
9
(2)9
− 9q 9/8 (18)
(1)3 (4)3
(9)3 (36)3
−8
23s (1 + 31−2s )L
−4
T(3.7)
23s (1 − 31−2s )L −8
T(3.8)
23s (1 + 32−2s )L
−4
T(3,9)
23s (1 − 32−2s )L
−8
T(3.10)
6.1 Three-dimensional sums
As a further example, consider k = 3 and l = 1. From (6.1.21),
∞
q 6n−5
q 6n−1
θ̄ (3, 1) 1/3
|q
=1+3
−
.
1 − q 6n−5
1 − q 6n−1
θ̄ (3, 1)
211
(6.1.23)
1
Now, the right-hand side of (6.1.23) is Entry (4.iii) in Chapter 19 of Ramanujan’s
Notebooks Part III [1] and is given in his notation. Translating this into
classical notation, it is just θ23 (q 1/2 )/4θ2 (q 3/2 ). Using (T1.3a) this becomes
(3)(2)6 /(1)3 (6)2 ; then, multiplying by (T1.6a) we obtain (T3.2). A sign transform
of (T3.2) yields (T3.1) and a Poisson transform of (T3.2) gives (T3.5). Finally a
sign transform of the latter gives (T3.6).
These six results appear to be the only ones in which members of can be
expressed as a single product of Eulerian terms. In attempting to find others using
(6.1.21), invariably a sum of two terms appears. Thus, for k = 6, l = 1,
∞
q 3n−2
q 3n−1
θ̄ (6, 1) 1/24
−1=6
−
.
|q
1 − q 3n−2
1 − q 3n−1
θ̄ (6, 1)
1
The Mellin transform of the right-hand side of the latter is 6L 1 (s)L −3 (s), which
is also the Mellin transform of θ3 θ3 (q 3 ) + θ2 θ2 (q 3 ) − 1 [27]. Hence
θ̄ (6, 1)|q 1/24 = θ̄ (6, 1)|q 1/24 θ3 θ3 (q 3 ) + θ2 θ2 (q 3 ) .
(6.1.24)
This is then the sum of two Eulerian products, and it does not seem possible
to collapse these into a single form. Furthermore, many different combinations
of the two terms can be made because many relations exist amongst theta functions. Thus the modular equation of order 3 may be expressed in terms of theta
functions as
θ3 θ3 (q 3 ) = θ2 θ2 (q 3 ) + θ4 θ4 (q 3 ).
So, (6.1.24) could be written as
θ̄ (6, 1)|q 1/24 = θ̄ (6, 1)|q 1/24 2θ2 θ2 (q 3 ) + θ4 θ4 (q 3 ) ,
resulting in a completely different combination of Eulerian terms. The simplest
form is obtained by inverting the Mellin transforms obtained from the expansions
of and X , and this is how (T3.7)–(T3.10) were obtained. It is noticeable that
the Mellin transforms of θ (3, l), θ̄ (3, 1), and θ (6, 1), which all contain L −3 ,
should be related but this is not evident from their Eulerian forms. However, it
may be simply demonstrated from two classical results for theta functions. The
first is θ3 = θ3 (q 4 ) + θ2 (q 4 ), which, if written in Eulerian form, is
(2)5
(8)5
(16)2
.
=
+ 2q
2
2
2
2
(8)
(1) (4)
(4) (16)
212
Closed-form evaluations of three- and four-dimensional sums
Multiplying both sides by q 1/3 (4)2 gives
q 1/3
5
2
(2)5
1/3 (8)
4/3 (4)(16)
,
=
q
+
2q
(8)
(1)2
(16)2
i.e.,
θ̄ (3, 1)|q 1/3 = θ (6, 1)|q 1/3 + 2θ (3, 1)|q 4/3 .
Similarly, from the classical result 2θ3 (q 4 ) = θ3 + θ4 we obtain
2θ̄ (6, 1)|q 1/3 = θ (3, 1)|q 1/3 + θ̄ (3, 1)|q 1/3 .
The results in Table 6.3 also have a long history. As previously mentioned
(T3.3) goes back to Jacobi. Results (T3.1) and (T3.5) are usually ascribed to [11].
However, they go back at least to Ramanujan [18], who quotes them along with
Jacobi’s result without reference, as though they were well known. However, formulae given by Ramanujan without comment imply several possibilities. It may
be that (a) he did not know a reference, (b) he thought the formulae too well
known to need a reference, (c) the formulae were his own, or (d) any combination
of the above. Thus (T3.1) and (T3.5) may well have originated with Ramanujan,
and up to now no earlier source has been found. Result (T3.2) is given by [14]
and is simply a sign transform of (T3.1). Result (T3.4) or its equivalent seems to
have appeared first in [10] and is simply a sign transform of (T3.3). Result (T3.6)
as presented here is new. It was suggested in [2] that performing the equivalent of
a sign transformation on what was essentially (6, 1) would yield an interesting
relation. The result is in fact φ (6, 1) and (T3.6). Results (T3.7)–(T3.10) are new.
(These remarks were made by Zucker [26]. Almost simultaneously a paper
was published by Köhler [15] in which (T3.1), (T3.2), and (T3.4)–(T3.6) were
given and derived in a completely different manner. Since Köhler’s paper was
received by his journal on 16 January 1989 and Zucker’s on 22 March 1989, Köhler should be credited with (T3.6). Both authors were in complete ignorance of one
another.)
It will be observed from Tables 6.1 and 6.3 that the members of and of X
contain w to a single power whereas the members of and of X contain w 3 . It is
thus possible to combine three suitable members of and X to form a members
of or of X . For example, remembering that x yz = 1,
θ (1, 0)θ̄ (1, 0)θ (4, 1)|q 1/4 = q 1/4 wz 2 wy 2 wx 2 = q 1/4 w 3 = θ (4, 1)|q 1/4 .
In classical notation this is Jacobi’s expression θ2 θ3 θ4 = θ1 . Written out fully
this is
2
2
2
2
(4n + 1)q (4n+1) /4 .
(−1)m q m +n +(4 p+1) /4 =
Now replacing q by e−t and taking Mellin transforms of both sides, we obtain
−s
= 4s L −4 (2s − 1). (6.1.25)
(−1)m m 2 + n 2 + (4 p + 1)2 /4
6.1 Three-dimensional sums
213
Thus the three-dimensional sum on the left-hand side of (6.1.25) has been
expressed in closed form, as was pointed out in [8]. Another example is
3
θ̄ (6, 1) = θ (4, 1)|q 3 , or
−s
(−1)m+n+ p (6m + 1)2 + (6n + 1)2 + (6 p + 1)2
= 3−s L −4 (2s − 1).
(6.1.26)
This result was given by [7]. It should be noticed that (6.1.26) is apparently
the first three-dimensional Coulomb sum to have been evaluated in closed
form and that for, s = 1/2, (6.1.26) is the first such analytic result of direct
physical significance (it is the electrostatic potential at six points in the unit cell
for NaCl). It is clear, then, that any combination of results in Tables 6.1 and 6.2
Table 6.4 Three-dimensional
sums evaluated in terms of a single Dirichlet
L-series. Here
implies summation over all indices m, n, p from −∞ to ∞;
A = L −4 (2s − 1), B = L −3 (2s − 1), C = (1 + 22−2s )L −3 (2s − 1),
D = (1 + 21−2s )L −3 (2s − 1), E = L −8 (2s − 1)
(−1)m [4m 2 + 4n 2 + (4 p + 1)2 ]−s = A
(−1)m [4m 2 + 4n 2 + (2 p + 1)2 ]−s = 2A
(−1)m+n [8m 2 + 8n 2 + (4 p + 1)2 ]−s = A
(−1)m+n [8m 2 + 8n 2 + (2 p + 1)2 ]−s = 2A
(−1)m+n+ p [8m 2 + 16n 2 + (4 p + 1)2 ]−s = A
(−1)m [8m 2 + (4n + 1)2 + (4 p + 1)2 ]−s = 2−s A
(−1)m [8m 2 + (4n + 1)2 + (2 p + 1)2 ]−s = 21−s A
(−1)m+n [(4m + 1)2 + (4n + 1)2 + 8 p 2 ]−s = 2−s A
(−1)m+n [16m 2 + (4n + 1)2 + (4 p + 1)2 ]−s = 2−s A
(−1)m+n [16m 2 + (4n + 1)2 + (2 p + 1)2 ]−s = 21−s A
(−1)m+n+ p [(6m + 1)2 + (6n + 1)2 + (6 p + 1)2 ]−s = 3−s A
(−1)m+n+ p [24m 2 + (6n + 1)2 + 2(6 p + 1)2 ]−s = 3−s A
(−1)m [(4m + 1)2 + (4n + 1)2 + 2(4 p + 1)2 ]−s = 2−2s A
(−1)m [(4m + 1)2 + (2n + 1)2 + 2(4 p + 1)2 ]−s = 21−2s A
(−1)m [(4m + 1)2 + (4n + 1)2 + 2(2 p + 1)2 ]−s = 21−2s A
(−1)m [(4m + 1)2 + (2n + 1)2 + 2(2 p + 1)2 ]−s = 22−2s A
(−1)m+n [(6m + 1)2 + 2(6n + 1)2 + 3(4 p + 1)2 ]−s = 6−s A
(−1)m+n [(6m + 1)2 + 2(6n + 1)2 + 3(2 p + 1)2 ]−s = 2 × 6−s A
(−1)m+n+ p( p+1)/2 [3(4m + 1)2 + 2(6n + 1)2 + (6 p + 1)2 ]−s = 6−s A
(−1)m+n+ p( p+1)/2 [(6m + 1)2 + 4(6n + 1)2 + (6 p + 1)2 ]−s = 6−s A
(−1)m [4(3m + 1)2 + 4(3n + 1)2 + (6 p + 1)2 ]−s = 9−s A
cont.
214
Closed-form evaluations of three- and four-dimensional sums
Table 6.4 (cont.)
(−1)m+n [72m 2 + (6n + 1)2 + 8(3 p + 1)2 ]−s = 9−s A
(−1)m [8(3m + 1)2 + 9(4n + 1)2 + (6 p + 1)2 ]−s = 18−s A
(−1)m+n(n−1)/2 [16(3m + 1)2 + (6n + 1)2 + (6 p + 1)2 ]−s = 18−s A
(−1)m+n(n−1)/2 [9(4m + 1)2 + (6n + 1)2 + 8(3 p + 1)2 ]−s = 18−s A
[2 − (−1)m+n ](−1) p [24m 2 + 72n 2 + (6 p + 1)2 ]−s = (1 + 31−2s )A
[1 + (−1)m+n ](−1) p [2m 2 + 6n 2 + (6 p + 1)2 ]−s = 2(1 + 32−2s )A
[2 − (−1)m+n ](−1) p [8m 2 + 24n 2 + (6 p + 1)2 ]−s = (1 + 32−2s )A
[1 + (−1)m+n ](−1) p [6m 2 + 18n 2 + (6 p + 1)2 ]−s = 2(1 + 31−2s )A
(−1)m cos 2π3m cos 2π3 n cos 2π3 p [4m 2 + 4n 2 + (2 p + 1)2 ]−s = 2−1 A
(−1)m+n cos 2π3 n cos 2π3 p [8m 2 + 8n 2 + (2 p + 1)2 ]−s = 2−1 A
(−1)m cos 2π3m cos 2π3 n [8m 2 + (2n + 1)2 + (4 p + 1)2 ]−s = 2−1−s A
(−1)m cos 2π3m cos 2π3 n [8m 2 + (2n + 1)2 + (2 p + 1)2 ]−s = 2−s A
(−1)m+n(n+1)/2 cos 2π3 n cos 2π3 p [(4m + 1)2 + (2n + 1)2 + 8 p 2 ]−s = 2−1−s A
(−1)m+n(n+1)/2 cos 2π3m cos 2π3 n cos 2π3 p [16m 2 + (2n + 1)2 + (2 p + 1)2 ]−s = 2−2−s A
(−1)m+n+ p [3m 2 + 12n 2 + (6 p + 1)2 ]−s = B
(−1)m+n+ p [6m 2 + (6n + 1)2 + (6 p + 1)2 ]−s = 2−s B
(−1)m+n [12m 2 + (6n + 1)2 + 3(4 p + 1)2 ]−s = 4−s B
(−1)m+n [12m 2 + (6n + 1)2 + 3(2 p + 1)2 ]−s = 2 × 4−s B
(−1)m+n+ p( p+1)/2 [12m 2 + 3(4n + 1)2 + (6 p + 1)2 ]−s = 4−s B
(−1)m+n [(6m + 1)2 + (6n + 1)2 + 6(4 p + 1)2 ]−s = 8−s B
(−1)m+n [(6m + 1)2 + (6n + 1)2 + 6(2 p + 1)2 ]−s = 2 × 8−s B
(−1)m+n+ p [3(4m + 1)2 + (4n + 1)2 + 2(6 p + 1)2 ]−s = 8−s B
(−1)m [(6m + 1)2 + 3(4n + 1)2 + 12(4 p + 1)2 ]−s = 16−s B
(−1)m [(6m + 1)2 + 3(4n + 1)2 + 12(2 p + 1)2 ]−s = 2 × 16−s B
(−1)m+n(n+1)/2+ p [3(4m + 1)2 + (6n + 1)2 + 12(4 p + 1)2 ]−s = 16−s B
(−1)m [(6m + 1)2 + 3(4n + 1)2 + 12(2 p + 1)2 ]−s = 2 × 16−s B
(−1)m+n(n+1)/2+ p [3(4m + 1)2 + (6n + 1)2 + 12(2 p + 1)2 ]−s = 2 × 16−s B
(−1)m [2(6m + 1)2 + 3(8n + 1)2 + 3(8 p + 1)2 ]−s = 32−s B
(−1)m+n+ p( p+1)/2 [3(8m + 1)2 + 3(8n + 3)2 + 2(6 p + 1)2 ]−s = 32−s B
(−1)m+n [12m 2 + (6n + 1)2 + 3 p 2 ]−s = C
(−1)m+n [(6m + 1)2 + (6n + 1)2 + 6 p 2 ]−s = 2−s C
(−1)m [(6m + 1)2 + 12n 2 + 3(4 p + 1)2 ]−s = 4−s C
(−1)m [(6m + 1)2 + 12n 2 + 3(2 p + 1)2 ]−s = 2 × 4−s C
(−1)m [2(6m + 1)2 + 3(4n + 1)2 + 3(4 p + 1)2 ]−s = 8−s C
(−1)m [2(6m + 1)2 + 3(4n + 1)2 + 3(2 p + 1)2 ]−s = 2 × 8−s C
(−1)m+n(n+1)/2 [4(6m + 1)2 + (6n + 1)2 + 3(4 p + 1)2 ]−s = 8−s C
(−1)m(m+1)/2+n(n+1)/2 [(6m + 1)2 + (6n + 1)2 + 6(4 p + 1)2 ]−s = 8−s C
cont.
6.1 Three-dimensional sums
215
Table 6.4 (cont.)
(−1)m(m+1)/2+n(n+1)/2 [(6m + 1)2 + (6n + 1)2 + 6(2 p + 1)2 ]−s = 2 × 8−s C
(−1)m+n(n+1)/2 [4(6m + 1)2 + (6n + 1)2 + 3(2 p + 1)2 ]−s = 2 × 8−s C
(−1)m+n+ p [24m 2 + 24n 2 + (6 p + 1)2 ]−s = D
(−1)m+n [12m 2 + (6n + 1)2 + 12 p 2 ]−s = D
(−1)m+n [(6m + 1)2 + (6n + 1)2 + 24 p 2 ]−s = 2−s D
(−1)m+n+ p( p+1)/2 [48m 2 + (6n + 1)2 + (6 p + 1)2 ]−s = 2−s D
(−1)m+n(n+1)/2+ p( p+1)/2 [24m 2 + (6n + 1)2 + (6 p + 1)2 ]−s = 2−s D
(−1)m [(6m + 1)2 + 48n 2 + 3(4 p + 1)2 ]−s = 4−s D
(−1)m [(6m + 1)2 + 48n 2 + 3(2 p + 1)2 ]−s = 2 × 4−s D
(−1)m+n(n+1)/2 [3(4m + 1)2 + (6n + 1)2 + 48 p 2 ]−s = 4−s D
(−1)m(m−1)/2+n(n+1)/2+ p( p+1)/2 [3(4m + 1)2 + 3(4n + 1)2 + 2(6 p + 1)2 ]−s = 8−s D
(−1)m [(4m + 1)2 + 8n 2 + 8 p 2 ]−s = E
(−1)m [16m 2 + 8n 2 + (4 p + 1)2 ]−s = E
(−1)m [16m 2 + 8n 2 + (2 p + 1)2 ]−s = 2E
(−1)m+n(n+1)/2 [2(6m + 1)2 + (6n + 1)2 + 24 p 2 ]−s = 3−s E
(−1)m(m+1)/2+n(n+1)/2+ p( p+1)/2 [(6m + 1)2 + (6n + 1)2 + (6 p + 1)2 ]−s = 3−s E
(−1)m(m−1)/2 [(6m + 1)2 + 72n 2 + 8(3 p + 1)2 ]−s = 9−s E
(−1)m(m−1)/2 cos 2π3m cos 2π3 n [(2m + 1)2 + 8n 2 + 8 p 2 ]−s = 2−1 E
2[(−1)m+n − 1](−1) p ( p + 1)/2[24m 2 + 72n 2 + (6 p + 1)2 ]−s ] = (1 − 31−2s )E
2[(−1)m+n − 1](−1) p ( p + 1)/2[8m 2 + 24n 2 + (6 p + 1)2 ]−s ] = (1 − 32−2s )E
(−1)m(m+1)/2 [(6m + 1)2 + 24n 2 + 24 p 2 ]−s = L −24 (2s − 1)
which yield a result in Table 6.4 will lead to the evaluation of a three-dimensional
sum in terms of a single Dirichlet L-series. In Table 6.4 a list of nearly 80 such
evaluations is given. It is not known whether this list is exhaustive or whether
any given result is a trivial consequence of another. Several results in Table 6.3
that are not immediately obvious from the entries in Tables 6.1 and 6.2 have
been obtained as sign or Poisson transforms of other results. For integer s all
these three-dimensional sums can be expressed
√ exactly in terms of powers of
π and surds, since L −d (2s − 1) = Rπ 2s−1 d, where R is a rational number;
see [27].
Although an attempt has been made to obtain these results in a consistent fashion, it is evident that complete success has not been achieved. The evaluation of
three-dimensional sums seems to depend on finding as a single term in Eulerian
form, and no systematic way of doing this is known or is necessarily available.
The numerical factors 24 also appear to play a central role in these matters. For
example, it is not possible to put θ (5, 1) into Eulerian form. Clearly the processes
216
Closed-form evaluations of three- and four-dimensional sums
described might be carried further. For example one could define
2
θ̄ (k, l) =
(kn + l)2 q (kn+l) .
There would be no difficulty in finding the Mellin transforms of such quantities.
Thus
2
2
2
2
2
(6n + 1)2 q (kn+1) = q 1 − 52 q 5 − 72 q 7 + 112 q 11 + · · ·
θ̄ (6, 1) =
and
M θ̄ (6, 1) = 1 − 52s−2 − 72s−2 + 112s−2 + · · · = L 12 (2s − 2),
and the Mellin transforms of members of and X obviously yield positiveparity L-series of order 2s − 2. However, the problem of obtaining as a single
Eulerian product seems even more intractable than that for , and so far no success with any has been achieved. If Eulerian or Jacobian expressions for could be obtained then certain five-dimensional sums could be found in closed
form, and the progression to higher odd dimensions is obvious. However, only
one five-dimensional sum, with a quadratic form but lacking the squared terms, is
known in closed form [9] (see Commentary 6.3 after the next section). This was
derived from a q-series based on manipulation of a basic hypergeometric series.
This method has not yet been explored further.
6.2 Four-dimensional sums
Shortly after the publication of the work in Chapter 1, the short paper [7]
appeared; this used the theta function method to obtain a variety of intriguing
sums. One is based on a q-series of Jacobi [13], which can be formulated as
∞
√
(−1)n (2n + 1)q n(n+1)/2 .
Q 30 ( q) =
(6.2.1)
n=0
By setting q = e−2t/3 , multiplying$ (6.2.1) on both sides by any inverse Laplace
∞
transform F[s], such that f (s) = 0 e−st F(t) dt, and integrating over s from 0
to ∞, one obtains the reduction formula
∞
(n + 12 )2
.
(−1)n 1 +n 2 +n 3 f || Rn + ( 16 , 16 , 16 )||2 =
(−1)n (2n + 1) f
3
Rn
n=0
(6.2.2)
(Here Rn = (n 1 , n 2 , n 3 ) and the sum is over all integer values −∞ < n i < ∞.)
1/2
In particular, for the choice f (s) = s −1/2 e−as , the sum on the right-hand side
of (6.2.2) is elementary and yields the Yukawa potential sum
√
e−a|| Rn +(1/6,1/6,1/6)||
a
=
.
(6.2.3)
(−1)n 1 +n 2 +n 3
3
sech
√
n + ( 1 , 1 , 1 )||
||
R
12
6
6
6
Rn
6.2 Four-dimensional sums
217
√
The value in the Coulomb limit a → 0+ is 3. This appears to be the known
value that is ‘closest’ to the Madelung constant and, other than at points where
the potential vanishes by symmetry, it remains the only Coulomb sum known
exactly for an actual crystal.
A second result worth quoting is the n-fold sum
∞
···
l1 =1
∞
(−1)l1 +···+ln
ln =1
ρn
(−1)n
= n
,
sinh πρn
2 (2n + 1)π
(6.2.4)
'
where ρn = l12 + · · · + ln2 . This is the n-dimensional analogue of the sum
(1.5.7), which was evaluated by contour integration. Next we examine a fourdimensional analogue of (6.2.3) and see that it has revealing connections with the
fascinating subject of elliptic curves and modular functions.
The four-dimensional analogue of the shifted rocksalt series (6.1.26) is
S(s) =
∞
(−1)n 1 +n 2 +n 3 +n 4
n j =−∞
[(n 1 + 16 )2 + (n 2 + 16 )2 + (n 3 + 16 )2 + (n 4 + 16 )2 ]s
.
(6.2.5)
Its evaluation takes us into new aspects of theta functions and q-series and even
displays a connection with Wiles’ famous resolution of Fermat’s last theorem. We
begin in the usual way by exponentiating the denominator to get
4
∞
∞
1
s−1
n −t (n+1/6)2
dt t
(−1) e
.
(6.2.6)
S(s) =
(s) 0
n=−∞
By Euler’s pentagonal number identity
∞
(−1)n q k(6n+1)
2 /24
= q k/24
n=−∞
∞
(1 − q kn ),
(6.2.7)
n=1
where q = e−2t/3 , one has
S(s) =
( 23 )s
(s)
∞
dt t s−1 η4
η(τ ) = e2iπ τ/24
∞
1
3
it
(6.2.8)
(1 − e2nπiτ )
(6.2.9)
0
where
n=1
is the Dedekind eta function. In passing we note that [17] η4 (6τ ) is the modular function associated with the elliptic curves of conductor 36, as required by
the restricted Taniyama–Shimura conjecture proved by A. Wiles. This will be
commented on below. By means of the Euler and Jacobi q-series given in [17]
for η and its cube, we find
∞
∞
∞ ( 2 )s 2
2
(−1)m+n (2m + 1)
dt t s−1 e−[(6n+1) +3(2m+1) ]/36 .
S(s) = 3
(s) n=−∞
0
m=0
(6.2.10)
218
Closed-form evaluations of three- and four-dimensional sums
Next, by evaluating the integral in (6.2.10) we obtain the double-series reduction
of (6.2.5):
S(s) = 12s
∞ ∞
(−1)m+n
n=−∞ m=0
[(6n
2m + 1
.
+ 3(2m + 1)2 ]s
+ 1)2
(6.2.11)
When s is an integer the m-series in (6.2.11) can be evaluated in terms of
hyperbolic functions, yielding a single-series reduction of S(s). Thus, we find
∞
∞
4π 4π (−1)n χ (n)
π
S(1) = √
(6.2.12)
(−1)n sech √ (6n +1) = √
√
6 n=−∞
6 n=1 cosh(nπ/ 12)
12
where χ (2n) = χ (3n) = 0 and χ (6n ± 1) = 1. Since
∞
(−1)(n+1)/2
√
cosh(π
n/
12)
n odd
equals K /k for a certain modulus, S(1) is elliptic in nature and, one expects,
has a hypergeometric representation. This has been proved explicitly for S(2) by
Rogers and Zudilin [21] and by Rodriguez-Villegas [19], who found that
4 5
,
,
1,
1
2π 2
; 12
.
(6.2.13)
ln 54 − 19 4 F3 3 3
S(2) =
81
2, 2, 2
At this point it is appropriate to summarize briefly several key mathematical
notions. An elliptic curve E, over the rational numbers, say, can be described by
an equation of the form y 2 = x 3 + ax + b with a, b ∈ Z and has a well-defined
tangent at each point (x, y). Then the discriminant D = 16(4a 3 + 27b2 ) = 0.
If E has rational points R, R , the line through R and R must intersect E at
some point (including the point O at infinity) (r1 , r2 ). Addition is defined by
R + R = (r1 , −r2 ) with respect to which the set of rational points forms an
abelian group, which, by the Mordell–Weil theorem, has the form Z r ⊕T . Furthermore, Mazur’s theorem states that the torsion group T must be either Z/mZ (m =
1, 2, . . . , 10 or 12) or Z/2Z ⊕ Z/2mZ (m = 1, 2, 3, 4). Determining the rank r
is complicated.
If the prime p does not divide D then, with respect to the finite field F p =
{0, 1, . . . , p − 1}, E will consist of ν p points and we define a p = 1 + p − ν p .
√
Hasse’s theorem states that |a p | ≤ 2 p. The Hasse–Weil L-function is defined by
−1 ∞
ap
an
1
1 − s + 2s−1
=
,
L(E, s) =
p
ns
p
p
(6.2.14)
n=1
where the second equality in (6.2.14) defines an for non-prime n. Three major
results are:
6.2 Four-dimensional sums
Weil–Wiles theorem
expression
219
There is an integer N (the conductor) such that the
(E, s) =
N
4π 2
s/2
(s)L(E, s)
continues analytically to an entire function of s satisfying the functional equation
(E, s) = ±(E, 1 − s).
Modularity conjecture
(6.2.15)
The function
f (z) =
∞
an einz
(6.2.16)
n=1
is modular.
Birch–Swinnerton-Dyer conjecture
We have
L(E, s) = O((s − 1)r )
as s → 1.
(6.2.17)
Wiles famously proved the modularity conjecture for certain conductors; values
and references can be found in [17]. A one-million-dollar prize still awaits a proof
of the third item. The conductor N is closely related to the product of the primes
dividing the discriminant, so is an invariant. From our discussion of S(s) we have,
for conductor N = 36, that the modular function f is η4 and therefore
L(E 36 , s) =
1
S(s).
9s
(6.2.18)
The Mahler measure of a multivariate polynomial P(x1 , x2 , . . . , xk ) is
defined by
d t ln |P(e2πit1 , . . . , e2πitk )|.
(6.2.19)
m(P) =
[0,1]k
The evaluation of the Mahler measures arising in our context in hypergeometric
form is quite non-trivial and has been carried out by Rodriguez-Villegas [19], and
Rogers and Zudilin [21]. In 1997 Deninger [6] conjectured that m(1 + x + 1/x +
y + 1/y) = (15/4π 2 )L(E 15 , 2). Subsequently Boyd [3], guided by numerical
results, conjectured over 100 such relations for various conductors, many lying
outside the class considered by Wiles and all for s = 2. In their seminal work
[21], Rogers and Zudilin put these in terms of the class of lattice sums
F(a, b, c, d) =
∞
(−1)n 1 +n 2 +n 3 +n 4 (a + b + c + d)2
n j =−∞
a(6n 1 + 1)2 + b(6n 2 + 1)2 + c(6n 3 + 1)2 + d(6n 4 + 1)2
2
.
(6.2.20)
220
Closed-form evaluations of three- and four-dimensional sums
By means of these connections, to date the following hypergeometric evaluations
have been proved:
4 5
, 3 , 1, 1 1
2π 2
1
3
; 2
F(1, 1, 1, 1) =
,
(6.2.21)
ln 54 − 9 4 F3
81
2, 2, 2
1 1 1
, ,
π2
F(1, 1, 2, 2) = √ 3 F2 2 2 3 2 ; 12 ,
1,
8 2
2
4 5
2
, 3 , 1, 1
4π
1
1
3
; −8
F(1, 1, 3, 3) =
.
ln 6 + 108 4 F3
81
2, 2, 2
(6.2.22)
(6.2.23)
Some examples of the conjectured and since proved (see e.g. [21]) evaluations
are:
4 5
,
,
1,
1
3π 2 5
1
3 3
; 27
ln 2 − 16
F(1, 1, 5, 5) =
,
(6.2.24)
4 F3
40 3
2, 2, 2 32
F(1, 2, 3, 6) =
π2
3 F2
12
F(1, 3, 5, 15) =
π2
3 F2
15
1 1 1
2, 2, 2
1, 32
1 1 1
2, 2, 2
1, 32
;
1
4
;
1
16
,
(6.2.25)
.
(6.2.26)
For further examples, details, and references see [20, 21]. It is clear that much of
this material is worthy of further investigation.
6.3 Commentary: A five-dimensional sum
Whereas many closed forms for lattice sums in two, four, six, and eight dimensions may be found – see [2, 23, 25] – results for odd dimensions are notoriously
difficult to establish. Indeed, in Chapter 1 it was noted that only two exact results
for three-dimensional lattice sums were known when the material in that chapter
was first published. That so many more three-dimensional sums have now been
found in closed form is possibly an aberration: none of them can be linked to any
physical lattice. It is noteworthy that the closed-form results for three-dimensional
lattice sums appear as a single L-series. For even dimensions one always find pairs
of products of two L-series. In five dimensions it seems only one result has been
established [9]. This is
∞
(m 1 m 2 + m 2 m 3 + m 3 m 4 + m 4 m 1 + m 2 m 5 )−s
m 1 ,m 4 =0
m 2 ,m 3 ,m 5 =1
= ζ (s)ζ (s − 2) − ζ 2 (s − 1),
and again it is expressed as sums of pairs of L-series.
(6.3.1)
6.4 Commentary: A functional equation for a three-dimensional sum 221
However, it is possible to find lattice sums which result in the product of four
L-series. Thus, if
2
m 2 + n 2 + p 2 + q 2 + mp + nq
ζ K (s) =
(m,n, p,q=0,0,0,0)
−3 (mq + np + nq)2
s
and
√
√
n + 12 p 3 + 12 q ≥ 0,
mq + np + nq ≥ 0,
m − p − n 3 > 0,
m 2 + n 2 + p 2 + q 2 + mp + nq − 3 (mq + np + nq) > 0
then
ζ K (s) = L 1 (s)L −3 (s)L −4 (s)L 12 (s).
This was found by Shail and Zucker (1985, unpublished). The summand in ζ K is
the norm of the numbers represented by the imaginary
bicyclic
√ biquadratic√field
√
formed by the adjunction of the four fields N, Q( −3), Q( −4), and Q( 12).
This particular field has class number 1. Details of such fields may be found in [5].
Brown and Parry [4] showed that there are just 47 imaginary bicyclic biquadratic
fields with class number 1; hence more results are available similar to the one
given above. Thanks are due to M. Peters (Wilhelms-Universität Münster) for
providing this information.
6.4 Commentary: A functional equation for a
three-dimensional sum
In Section 1.2, and again in Commentary 3.9, the Mellin transform, the Poisson
summation formula, and Hobson’s integral were used together to write some lattice sums as rapidly convergent series. Here, we use this technique to re-derive
the functional equation (1.3.42) for a three-dimensional lattice sum
%
&−s
m 2 + n 2 + p2 .
a(2s) =
m,n, p
First, the Mellin transform gives
a(2s) =
1 ∞ s−1 −t (m 2 +n 2 + p2 )
t e
dt.
(s) m,n, p 0
We can then proceed in two ways: apply the Poisson summation formula to a
single sum, say, over m, or apply it twice to a pair of sums – say over n and p.
Thus, we produce
222
Closed-form evaluations of three- and four-dimensional sums
a(2s) =
=
√
1
π
+
2s
(s)
m
m
n, p
m; n, p=0 0
1
π
+
(s)
(n 2 + p 2 )s
∞
t s−3/2 e−t (n
∞
2 + p 2 )−π 2 m 2 /t
t s−2 e−tm
dt
2 −π 2 (n 2 + p 2 )/t
(6.4.1)
dt.
m=0; n, p 0
(6.4.2)
Using Hobson’s integral
∞
q s/2
√
t s−1 e− pt−q/t dt = 2
K s (2 pq),
p
0
which is valid for Re p, Re q > 0, we can develop the above expressions into
triple sums over the modified Bessel function of the second kind; terms which
cannot be evaluated using Hobson’s integral are treated separately, resulting in
line and plane sums.
Evaluating the plane sums using the result
%
&−s
m 2 + n2
= 4ζ (s)L −4 (s),
C(s) :=
m,n
equations (6.4.1) and (6.4.2) reduce to the expressions
√
π (s − 12 )
a(2s) = 2ζ (2s) +
C(s − 12 )
(s)
s−1/2
∞
'
%
&
m
4π s K s−1/2 2π m n 2 + p 2 , (6.4.3)
+
(s)
n 2 + p2
m=1 n, p
a(2s) =
2π
ζ (2s − 2) + C(s)
s−1
2
∞
'
%
&
4π s n + p 2 s−1
+
K s−1 2π m n 2 + p 2 .
(s)
m
n, p
(6.4.4)
m=1
Consider now the transformation s → 32 − s applied to (6.4.3): the summand
in (6.4.3) transforms into the summand in (6.4.4), since K ν (z) = K −ν (z). Substituting the transformed equation into (6.4.4), we obtain a relation between a(2s)
and a(3 − 2s). This relation can be simplified using the reflection formulae of the
ζ, L −4 , and functions and the duplication formula of the latter; in the end we
obtain
a(3 − 2s)( 32 − s)
a(2s)(s)
,
(6.4.5)
=
πs
π 3/2−s
which is equation (1.3.42) and provides an analytic continuation for a(2s).
Note that this process can be generalized to higher dimensions, and it provides
a rich source of sum equalities for K s ; the equality of (6.4.3) and (6.4.4) is but
one example. Similar equations are derivable for alternating sums.
6.5 Commentary: Two amusing lattice sum identities
223
6.5 Commentary: Two amusing lattice
sum identities
(1) Show that, for k = 0,
∞
(−1)
n
n=0
sinh πk (2n + 1)
cosh2 πk (2n + 1) − cos2
πl
k
∞
(−1)n
k
πl = sec
.
4
k n=−∞ cosh π2 (kn + l)
Proof We start with the double sum
∞
∞ (−1)m+n
n=−∞ m=0
2m + 1
.
(kn + l)2 + (2m + 1)2
The n-summation is tabulated in [7]:
sinh πk (2m + 1) cos πl
π
k
k(2m + 1) cosh2 πk (2m + 1) − cos2
πl
k
.
Similarly, the m-summation is
π
π
sech (kn + l).
4
2
Hence, equating the two remaining sums we have the stated result.
(2) Another amusing evaluation, which can be traced back to Liouville, is
∞
(n 2
n 1 ,n 2 ,n 3 ,n 4 =0 1
+ n 22
+ n 23
+ n 24
1
+ n 1 + n 2 + n 3 + n 4 + 1)s
= (1 − 2−s )(1 − 21−s )ζ (s)ζ (s − 1),
valid for Re s > 2.
Proof The sum can be written as
∞
n 1 ,n 2 ,n 3 ,n 4
∞
= Ms
1
%
&s
1 2
1 2
(n 1 + 2 ) + (n 2 + 2 ) + (n 3 + 12 )2 + (n 4 + 12 )2
=0
q
(n 1 +1/2)2
n 1 =0
=
4
1
2
∞
n 2 =0
q
(n 2 +1/2)2
∞
n 3 =0
q
(n 3 +1/2)2
∞
q
(n 4 +1/2)2
n 4 =0
Ms [θ24 ],
where as usual Ms denotes the Mellin transform. Now the result follows; see [23].
224
Closed-form evaluations of three- and four-dimensional sums
References
[1] B. C. Berndt. Ramanujan’s Notebooks Part III. Springer-Verlag, New York, 1991.
[2] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number
Theory and Computational Complexity. Wiley, New York, 1987.
[3] D. W. Boyd. Mahler’s measure and special values of L-functions. Exp. Math., 7:37–
82, 1998.
[4] E. Brown and C. J. Parry. The imaginary bicyclic biquadratic fields with class number
1. J. Reine Angew. Math., 266:118–120, 1974.
[5] H. Cohn. A Classical Invitation to Algebraic Numbers and Class Fields. SpringerVerlag, Berlin and New York, 1978.
[6] C. Deninger. Deligne periods of mixed motives, K -theory and the entropy of certain
Z n -actions. J. Amer. Math. Soc., 10:259–281, 1997.
[7] P. J. Forrester and M. L. Glasser. Some new lattice sums including an exact result for
the electrostatic potential within the NaCl lattice. J. Phys. A, 15:911–914, 1982.
[8] M. L. Glasser. The evaluation of lattice sums. I. Analytic procedures. J. Math. Phys.,
14:409–413, 1973.
[9] M. L. Glasser. The evaluation of lattice sums. IV. A five-dimensional sum. J. Math.
Phys., 16:1237–1238, 1975.
[10] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and
Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980.
[11] B. Gordon. Some identities in combinatorial analysis. Quart. J. Math., 12:285–290,
1961.
[12] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers, 4th
edition. Clarendon, Oxford, 1960.
[13] C. G. Jacobi. Fundamenta Nova Theoriae Functionum Ellipticarum. Konigsberg,
1829.
[14] V. G. Kac. Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius
function and the very strange formula. Adv. Math., 30:85–136, 1978.
[15] G. Köhler. Some eta-identities arising from theta series. Math. Scand., 66:147–154,
1990.
[16] L. Lorenz. Bidrag tiltalienes theori. Tidsskrift Math., 1:97–114, 1871.
[17] Y. Martin and K. Ono. Eta-quotients and elliptic curves. Proc. Amer. Math. Soc.,
125:3169–3176, 1997.
[18] S. Ramanujan. On certain arithmetical functions. Trans. Camb. Phil. Soc., 22:159–
184, 1916.
[19] F. Rodriguez-Villegas. Modular Mahler measures I. In Topics in Number Theory,
pp. 17–48. Kluwer, Dordrecht, 1999.
[20] M. Rogers. Hypergeometric formulas for lattice sums and Mahler measures. IMRN,
17:4027–4058, 2011.
[21] M. Rogers and W. Zudilin. From L-series of elliptic curves to Mahler measures.
Compositio Math., 148:385–414, 2012.
[22] H. J. S. Smith. Report on the Theory of Numbers. Chelsea, New York, 1865.
[23] I. J. Zucker. Exact results for some lattice sums in 2, 4, 6 and 8 dimensions. J. Phys.
A, 7:1568–1575, 1974.
References
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[24] I. J. Zucker. New Jacobian θ functions and the evaluation of lattice sums. J. Math.
Phys., 16:2189–2191, 1975.
[25] I. J. Zucker. Some infinite series of exponential and hyperbolic functions. SIAM J.
Math. Anal., 15:406–413, 1984.
[26] I. J. Zucker. Further relations amongst infinite series and products II. The evaluation
of 3-dimensional lattice sums. J. Phys. A, 23:117–132, 1990.
[27] I. J. Zucker and M. M. Robertson. Systematic approach to the evaluation of
2
2 −s
(m,n=0,0) (am + bmn + cn ) . J. Phys. A, 9:1215–1225, 1976.
7
Electron sums
In 1934 Wigner [17] introduced the concept of an electron gas bathed in a
compensating background of positive charge as a model for a metal. He suggested that under certain circumstances the electrons would arrange themselves
in a lattice, and that the body-centred lattice would be the most stable of the three
common cubic structures. Fuchs [14] appears to have confirmed this in a calculation on copper relying on physical properties of copper. The evaluation of the
energy of the three cubic electron lattices under precise conditions was carried
out by Coldwell-Horsfall and Maradudin [10] and became the standard form for
calculating the energy of static electron lattices, U (lattice). In this jellium model,
electrons are assumed to be negative point charges located on their lattice sites
and surrounded by an equal amount of positive charge uniformly distributed over
a cube centred at the lattice point. First, one calculates the interaction energy,
U1 , of a single electron with all the other electrons on their lattice sites. Then
one finds the energy of interaction of an electron with the compensating positive
background, U2 . Thus
U (lattice) = U1 − U2 .
(7.0.1)
The procedure is outlined here with the simple cubic lattice as its paradigm.
Essentially one attempts to evaluate (7.0.1), where
U1 =
1
e 2 ,
a0
(m 2 + n 2 + p 2 )1/2
U2 =
e2
a0
∞ ∞ ∞
0
0
0
d x d y dz
.
(x 2 + y 2 + z 2 )1/2
(7.0.2)
In (7.0.2) e is the charge on an electron, a0 is the side of the elementary cube of
the lattice, and is the sum over all integers m, n, p from −∞ to ∞ excluding
the case when m, n, p are simultaneously zero. Now both U1 and U2 are divergent
quantities, so the approach is to find ways of evaluating them so that the infinities
Electron sums
227
cancel. An outline of this procedure is given now; the common factor e2 /a0 is
ignored. The energy U1 is first expressed as an integral; thus
∞
U1 =
t −1/2 exp −π(m 2 + n 2 + p 2 )t dt.
0
Next, we break the range of integration into two parts, (0,1) and (1, ∞), and define
∞
U11 =
t −1/2 exp −π(m 2 + n 2 + p 2 )t dt,
U12 =
1
1
t −1/2 exp −π(m 2 + n 2 + p 2 )t dt.
0
In terms of the auxiliary integrals φk , which are given by
∞
φk (x) =
t k e−xt dt,
(7.0.3)
1
U11 can be written as follows:
φ−1/2 π(m 2 + n 2 + p 2 ) .
U11 =
(7.0.4)
For U12 , remove the restriction on the sum by subtracting the m = n = p = 0
term, thus obtaining
1
1
−1/2
2
2
2
t
exp −π(m + n + p )t dt −
t −1/2 dt.
(7.0.5)
U12 =
0
0
In (7.0.5) use the Poisson transform
∞
exp(−π m 2 t) =
−∞
∞
1
−π m 2
√ exp
t
t
−∞
and perform the second integral, to obtain
1
π
t −2 exp − (m 2 + n 2 + p 2 ) dt − 2.
U12 =
t
0
(7.0.6)
In (7.0.6) restore the restriction on the sum by adding back the m = n = p = 0
term and, in the remaining integral, substitute t = 1/u to obtain
1
∞
dt
exp −π u(m 2 + n 2 + p 2 ) du − 2 +
.
(7.0.7)
U12 =
2
1
0 t
On adding U11 and U12 we get
U1 =
φ−1/2 π(m 2 + n 2 + p 2 ) + φ0 π(m 2 + n 2 + p 2 ) − 2 +
0
where, of course, the integral is divergent.
1
1
dt,
t2
(7.0.8)
228
Electron sums
In a similar fashion U2 can be expressed as the following four-fold integral:
∞ ∞ ∞ ∞
U2 =
t −1/2 exp −π(x 2 + y 2 + z 2 )t dt d x dy dz. (7.0.9)
0
0
0
0
Changing to polar coordinates and performing the angular integrals leads to
∞ ∞
∞
1
dt
dt
U2 = 4π
t −1/2 exp(−π tr 2 )r 2 dr dt =
=
1
+
,
2
2
t
0
0
0
0 t
(7.0.10)
where the integral is again divergent. Subtract (7.0.10) from (7.0.8) and ‘cancel’
the identical divergent integrals; then we get, for the simple cubic lattice,
φ−1/2 π(m 2 + n 2 + p 2 ) +φ0 π(m 2 + n 2 + p 2 ) −3. (7.0.11)
U (SC) =
Tables of φk (x) were prepared by Born and Misra [4]. One only needs a few
terms of the series, since φk (x) rapidly becomes smaller as x increases. Thus
Coldwell-Horsfall and Maradudin [10] found that
U (SC) = −2.837297
e2
.
a0
However, if we look back to Section 1.3, where a(2s) = 1/(m 2 + n 2 + p 2 )2s
was treated as the three-dimensional analogue of the Riemann zeta function, we
see that U1 (SC) = a(1) and, by making use of the functional equation coming from analytic continuation, a(1) may be evaluated and is given in Table 1.5.
To much surprise it was seen that U1 (SC) found for a(1) by the simple method
described in Section 1.3 is equal to the expression U (SC) found by the rather
complex method given above. Results for electrons forming face-centred cubic
(FCC) and body-centred cubic (BCC) lattices are also found in [10]. It is simple
to show that for these lattices the interaction amongst the electrons alone is
U1 (FCC) = a(1) + d(1),
U1 (BCC) =
1
2
[a(1) + 3c(1)] ,
and it is equally simple to evaluate them; once again they were found to be equal
to the values in [10]. That is,
e2
e2
U1 (FCC) = U (FCC) = −4.584875 , U1 (BCC) = U (BCC) = −3.639240 .
a0
a0
To find the most stable of these lattices the results have to be expressed in terms
of rs , the radius of a sphere of volume equal to the volume per electron; we have
a03 = 43 πrs3 (SC),
1 3
4 a0
and thus
U (SC) = −1.760119
= 43 πrs3 (FCC),
e2
,
rs
1 3
2 a0
= 43 πrs3 (BCC),
U (FCC) = −1.791753
U (BCC) = −1.791860
e2
,
rs
e2
,
rs
Electron sums
229
in which the BCC lattice appears as the most stable.
Results obtained by Bonsall and Maradudin [3] by the traditional method for
two-dimensional electron lattices also agreed with closed-form results found by
direct evaluation of just the appropriate U1 . Thus for the square and triangular
lattices (Bonsall and Maradudin call the latter the hexagonal lattice) we have
U1 (sq) =
(m 2 + n 2 )−1/2 ,
U1 (tri) =
(m 2 + mn + n 2 )−1/2 .
A standard decomposition of these double sums into products of single sums, to
be found in Zucker and Robertson [19] and Borwein and Borwein [8], gives
U1 (sq) = 4ζ ( 12 )L −4 ( 12 ) = −3.900 264 924,
U1 (tri) = 6ζ ( 12 )L −3 ( 12 ) = −4.213 422 363
in terms of a0 , the side of the square or of the triangle, and it is very simple to
calculate these quantities. In terms of rs , the radius of a circle equal in area to the
area per electron, we have
a02 = πrs2 (sq),
√
3 2
2 a0
= πrs2 (tri),
so that
U (sq) =
−2.200488843e2
,
rs
U (tri) =
−2.212205173e2
.
rs
There were two surprises here. First these numbers could be obtained without
going through the long process just described, in particular, avoiding subtracting
two infinite integrals. Secondly, why was the actual energy of the lattice given just
by the interaction of the electrons alone, calculated in this way? Since it is much
easier to evaluate the analytic continuations of the corresponding lattice sums
without bothering to subtract divergent integrals ‘in the right way’, the question
arises whether using this method is valid for all Coulomb lattices. An attempt to
justify this procedure was explored by Borwein et al. [6]. They started by observing that the previous manipulations were not arbitrary, but rather were stable since
any answer might be obtained by inappropriate processes. Some regular limiting
procedure must be undertaken to guarantee a robust answer. The following principle is proposed: the rearrangements used should depend only on the geometry
of the underlying lattice and not on the power s in the law of interaction.
The consequence of this principle is that we look for an appropriate analytic
function for U (lattice : s) and take our answer the value of this function at
s = 12 . They argue that this forces the answer to be alat (1) where alat (2s) is the
d-dimensional sum over the lattice sites:
(l12 + l22 + · · · + ld2 )−2s ,
alat (2s) =
in which the sum represents alat (2s) for Re s > 12 d. This has an analytic continuation for 0 < Re s < 12 d, which, for the d-dimensional simple cubic lattice, is
230
Electron sums
obtained via the functional relation
π −s (s)alat (2s) = π s−d/2 1
d − s alat (d − 2s).
2
Formally, U (lattice : s) = alat (2s) − U2 (s), where
d x1 · · · d xd
...
U2 = c
% 2
&s ,
x1 + x22 . . . + xd2
()
while c is a constant appropriate to the lattice and is some volume in
d-dimensional space which gives electrical neutrality. Now split U2 into the
regions inside the unit sphere and outside the unit sphere thus:
U2 (s) = U2< (s) + U2> (s).
Adding on the finite integral U2< (s) when 0 < Re s < 12 d to both sides, we have
U (lattice : s) + U2< (s) = alat (2s) − U2> (s).
(7.0.12)
Now, U2< (s) may be integrated to give for Re s > 12 d
U2< (s) =
C
d − 2s
with
C=
c 2π d/2
( 12 d)
,
whereas for Re s > d/2 the other integral, U2> , is finite and, by direct computation
or by appealing to the central symmetry of the integral, has the value C/(2s − d).
Now we argue as follows. The left-hand side of (7.0.12), namely U (lattice : s) +
C/(d − 2s), is an analytic continuation of the right-hand side of (7.0.12) for Re
s < 12 d. Thus, by comparing the two we have
U (lattice : s) = alat (2s)
for Re s < 12 d and hence for s = 12 we have our result. By considering a
precise limiting process we shall demonstrate why a unique ‘answer’ is to be
obtained.
The following model in d dimensions will be considered. In this model
point charges are located at lattice sites and these are surrounded by an equal
amount of opposite charge uniformly distributed over hypercubes centred at the
lattice point and of side equal to the lattice spacing. This is illustrated there
after in two dimensions, where the shaded portion represents positive charge
of value equal to the point negative charge but uniformly distributed over a
square.
Electron sums
231
We shall thus examine the precise limiting procedure
N
N −s
lim σ N (s) = lim
l12 + l22 + · · · + ld2
···
N →∞
N →∞
−
−N
N +1/2
−N +1/2
···
−N
N +1/2
−N +1/2
x12 + x22 + · · · + xd2
−s
d x1 d x2 · · · d xd
= lim [α N (s) − β N (s)] ,
N →∞
although a priori this limit need not exist. This procedure maintains electrical neutrality throughout the limiting process. Further the model has zero dipole moment
and, in three dimensions, zero quadrupole moment as well.
Consider first the multidimensional integral β N (s) . Writing x j = (N + 12 )X j ,
we have
1
1
β N (s)
2
2
X
=
I
(s)
=
·
·
·
+
·
·
·
+
X
d d X 1 · · · d X d . (7.0.13)
1
(N + 12 )d−2s
−1
−1
Now, I (s) may be rewritten as follows:
d X1 · · · d Xd
I (s) = 2d · · ·
,
2
2 s
Wd (X 1 + · · · + X d )
where Wd is the pyramidal region |X i | ≤ X d ≤ 1. Making the substitution X i =
X d xi , we have
1
1
1
d x1 · · · d xd−1
2d
d−1−2s
C(s),
I (s) = 2d
Xd
d Xd
···
=
2
2
s
d
−
2s
0
−1
−1 (1 + x 1 + · · · x d−1 )
where
C(s) =
1
−1
···
1
−1
d x1 · · · d xd−1
,
2 )s
(1 + x12 + · · · xd−1
and this integral clearly converges for Re s > 0. Thus
σ N (s) = α N (s) − (N + 12 )d−2s
2d
C(s),
d − 2s
(7.0.14)
and the whole of the right-hand side of (7.0.14) is meromorphic for Re s > 0,
with a single simple pole at s = 12 d. Note that
β0 (s) = 22s−d
2dC(s)
(d − 2s)
(7.0.15)
gives an analytic continuation of the integral inside the unit hypercube. For
Re s > 12 d, σ N (s) is infinite since its defining integral is infinite. Then, for
Re s > 12 d,
lim σ N (s) + β0 (s) = lim α N (s) + β0 (s) = alat (2s) + β0 (s),
N →∞
N →∞
(7.0.16)
232
Electron sums
since β N (s) tends to zero. We can use (7.0.15) to see that alat (2s) + β0 (s) is analytic in Re s > 0. The principle of analytic continuation allows us to conclude that
(7.0.16) continues to hold in any half-plane in which the left-hand side exists and
is known to be analytic. However, for 0 < Re s < 12 d, β0 (s) is a finite integral.
Thus, in the appropriate strip, lim N →∞ σ N (s) = a1at (2s) for the particular limiting process we have undertaken – if it is, in fact, true that lim N →∞ σ N (s) + β0 (s)
exists and is analytic in the appropriate half-plane (or at least for 12 < Re s < 12 d
with continuity at 12 ).
Now, for all lattices in two dimensions, σ N (s) + β0 (s) can be shown to tend
to an analytic limit in the right half-plane. This was proved for the square lattice
in Borwein et al. [6]. While it might seem reasonable to presume that it holds
generally, that is in fact false. Considerations similar to those given there show
that, for the simple cubic lattice in three dimensions, the limit is analytic for 12 <
Re s < 32 but is discontinuous at 12 . Indeed lims→1/2+ lim N →∞ σ N (s) = alat (2s),
the correct answer, but this differs by π/6 from lim N →∞ σ N ( 12 ) (the rectangular
limit). This is discussed more fully in a further paper by Borwein et al. [5].
In addition, in two dimensions one can also show that the limit over expanding
circles, namely the limit as N → ∞ of (m 2 + n 2 )−s −
(x 2 + y 2 )−s d x d y
τ N (s) =
1≤(m 2 +n 2 )≤N
1≤(m 2 +n 2 )≤N
is analytic for Re s > 13 . As a consequence, when the integral inside the unit circle
is reintroduced, the corresponding limiting value at s = 12 is also 4ζ ( 12 )L −4 ( 12 ).
Granted the general applicability of the previous meta-principle – which we
have illustrated already for SC, FCC, BCC, square, and triangular lamina lattices –
we can easily determine U (lattice) for many other lattices. Thus other threedimensional lattices were investigated; in particular, various types of hexagonal
lattice were considered. A simple hexagonal lattice is a structure formed by stacking planes of two-dimensional triangular lattices directly above each other. The
direction of stacking is known as the c axis and the separation of planes in terms
of the nearest neighbour distance, R, in the triangular lattice is called the axial
ratio, c. If particles on such a lattice interact with an r −s potential, it is simple to
show that the appropriate lattice sums are given by
−s
(m − 12 )2 + 3(n − 12 )2 + c2 p 2
(m 2 +3n 2 +c2 p 2 )−s +
.
H (2s : c) =
Using our principle that the energy of a such a lattice of electrons in a compensating positive background is given in terms of e2 /R by
H (hex) = H (1 : c),
methods of converting such sums into rapidly converging cosech sums have been
previously described in Section 1.3 and, after some detailed algebra, H (1 : c) can
be converted to rapidly converging sums and double sums of cosech functions:
Electron sums
233
H (1 : c)
√
√ ∞
16 log 2
3π c
3 cosech(cπ m)
−
+
2
=−
3c
18
3
m
1
√
√
∞ √
∞
∞ 3 cosech(cπ m/ 3)
3 cosech cπ m 2 + n 2 /3
+2
+4
√
m
3m 2 + n 2
1
1
1
√
∞
∞ ∞
4 3
cosech(2π m/c)
4 cosech π 4m 2 /3 + 4n 2 /c2
+
2
+4
3c
2m/c
4m 2 /3 + 4n 2 /c2
1
1
1
√
√
∞
cosech( 12π m/c)
4 3
+
2
√
c
12m/c
1
∞ ∞
4 cosech π 12m 2 /3 + 12n 2 /c2
+4
12m 2 + 12n 2 /c2
1
1
√ ∞
√
3 (−1)m cosech( 3π m/c)
−
2
√
c
3m/c
1
∞
∞
(−1)m+n cosech π (3m 2 /3 + 3n 2 /c2 ) .
(7.0.17)
+4
3m 2 + 3n 2 /c2
1
1
Evaluations of H (1 : c) for several values of c are given below. The only
corresponding results we'know in the literature are those given by Hund [15,
16] for c = 2 and c = 83 . The latter value for c is the so-called ideal ratio.
Hund’s values were calculated by the traditional Ewald method and the numerical
accuracy was low. He gave
' H (1 : 2) = −1.796.
H 1 : 83 = −2.238,
As in the case of the three cubic lattices, we convert the values in terms of e2 /R
into values in
of e2 /rs . For the simple hexagonal lattice the volume of a
√terms
3
unit
√ cell is 3 3c R /2, and with three electrons per cell the volume per electron
is 3c R 3 /2. Hence
√
√
3c R 3 /2 = 4πrs3 /3
and
rs /R = (3 3c/8π )1/3 .
√
Multiplying H (1 : c) by (3 3c/8π )1/3 yields U (hex) in terms of e2 /rs , as in
Table 7.1. It is so simple to evaluate H (1 : c) via (7.0.17) that numerically the
minimum was located at c = 0.9284 with
U (hex)
= −1.774642655.
e2 /rs
We have also evaluated U for the hexagonal close-packed (HCP) structure,
which may be regarded as two equal interpenetrating hexagonal crystals with ideal
234
Electron sums
Table 7.1 Values for U (hex)
c2
U (hex)/(e2 /R)
U (hex)/(e2 /rs )
1
4
9
16
3
4
−3.263230504
−1.531505030
−3.253618321
−1.747971578
−3.143633242
−2.995711953
−2.730799747
−2.502936140
−2.238722127
−1.795017494
−1.771826683
−1.771389474
−1.727636634
−1.661251716
−1.558866728
−1.337291317
1
3
2
2
8
3
4
axial ratios, one lattice based at the origin (0, 0, 0) and the other at
lattice sums for such a structure in terms of R may be written
' HCP(2s) = H 2s : 83 + H ∗ (2s),
1 1 1
2, 3, 2
. The
where
H ∗ (2s) =
−s
(m − 12 )2 + 3(n − 13 )2 + 83 ( p − 12 )2
−s
+
m 2 + 3(n − 16 )2 + 83 ( p − 12 )2
.
√
For the HCP structure the volume of a hexagonal cell is 3 2R 3 , but there are
six electrons per cell. Hence we have
√ 3
√
2R /2 = 4πrs3 /3
and
rs /R = (3 2/8π )1/3 ,
and we obtain
U (HCP) = −3.241858662e2 /R = −1.791676267e2 /rs .
Foldy [12] found by the traditional method that U (HCP) = −1.79167624e2 /rs
and noted that if the axial ratio is 1.0016 times the ideal value then the last two
figures read 90 instead of 24, and he claims this is the real minimum.
As a final example, the result for the diamond lattice is given. The diamond
lattice may be considered as two equal interpenetrating FCC lattices, one based
on (0, 0, 0) and the other on ( 14 , 14 , 14 ). In terms of the cube side of an FCC lattice,
the diamond lattice sums are easily shown to be
di(2s) = FCC(2s) + 22s−1 [BCC(2s) − SC(2s)] ,
Electron sums
235
and again the principle gives U (di) = di(l) in terms of e2 /a0 . Using previous
results,
di(1) = 12 [a(1) + 3c(1) + 2d(1)] = −5.386789045.
For the diamond lattice, the volume of a cubic cell is U and this has eight electrons
per cell. Hence a03 /8 = 4πrs3 /3 and rs /a0 = (3/32π )1/3 , and we obtain
U (di) = −1.670851406e/rs ,
in complete agreement with Foldy’s 1978 value, obtained conventionally.
It should, however, be pointed out that in all the previous calculations the
structures were either Bravais lattices or contained a basis of energy equivalent
sites. With more complex lattices where this is not the case, ignoring this point
has led to incorrect results for the fluorite, perovskite, and spinel lattices – see
Zucker [18]. This was noted by Cockayne [9], who found the correct results
by the conventional Ewald approach. Baldereschi et al. [2] showed that by correctly superimposing the component Bravais lattices with the right weightings, the
method used here gives the correct values. Thus the perovskite lattice is a simple
cubic lattice with a five-fold basis, with the general formula AB X 3 . The A sites
are the corners of the cube, the B sites the centres of the cube and the X sites the
faces of the cube. The X sites, though energetically equivalent to each other, are
not equivalent to the A and B sites. One has to evaluate the interaction of each site
with the others and weight them accordingly. Thus the perovskite lattice may be
taken to be made up of the A lattice, an SC lattice based at the origin (0, 0, 0), the
B lattice, an SC lattice based on ( 12 , 12 , 12 ), and three X lattices, i.e., SC lattices
based on the equivalent sites (0, 12 , 12 ), ( 12 , 0, 12 ), and ( 12 , 12 , 0). So, the interaction
of A with the other sites may be written, in the notation of Section 1.3, as
ψ A = ψ(0, 0, 0) + ψ( 12 , 12 , 12 ) + 3ψ(0, 12 , 12 ) = a(1) + 12 [3c(1) − a(1)] + d(1).
Similarly, we have
ψ B = ψ(0, 0, 0) + ψ( 12 , 12 , 12 ) + 3ψ(0, 0, 12 ) = a(1) + 12 [3c(1) − a(1)]
+ 12 [3b(1) − d(1)],
and
ψ X = ψ(0, 0, 0) + 3ψ(0, 12 , 12 ) + ψ(0, 0, 12 ) = a(1) + f (1) + 16 [3b(1) − d(1)].
To obtain the correct electronic interaction we require to weight these terms
correctly:
UP =
1
5
(ψ A + ψ B + 3ψ X ) =
1
5
e2
7b(1) + 7c(1) + 13 13d(1) = 4.671242310 .
a0
236
Electron sums
Since there are five electrons per cube, in terms of rs this becomes
1.694 648 083e2 /rs , in complete agreement with Cockayne’s result. Similarly, for
the fluorite lattice AB2 , A is an FCC lattice based on (0, 0, 0) and the two B
lattices are FCC lattices based on the equivalent sites ( 14 , 14 , 14 ) and ( 34 , 34 , 34 ).As
before,
ψ A = ψ FCC (0, 0, 0) + 2ψ FCC 14 , 14 , 14 = a(1) + d(1) + 3c(1) − a(1)
and
ψ B = ψ FCC (0, 0, 0) + ψ FCC
1 1 1
4, 4, 4
+ ψ FCC
1 1 1
2, 2, 2
= a(1) + d(1) + 12 [3c(1) − a(1)] + a(1) − d(1).
Hence, after some further manipulation, we obtain
UFluorite =
1
12
(4ψ A + 8ψ B ) =
1
3
2
[3b(1) + 9c(1) + 2d(1)] = 6.380598623 ae 0 .
Since there are 12 electrons per cube, in terms of rs this becomes
1.728906369e2 /rs , again in complete agreement with Cockayne’s result. Of all
the three-dimensional structures considered, the BCC remains energetically the
most stable.
7.1 Commentary: Wigner sums as limits
We now describe an analysis from [5], as follows. The authors investigated the
limit as N → ∞ of the d-dimensional quantity
σ N (s) := α N (s) − β N (s),
where
α N (s) =
β N (s) =
N
···
N
−N
−N
N +1/2
−(N +1/2
s
f (n 1 , n 2 , . . . , n d ) ,
···
N +1/2
−(N +1/2
f (x1 , x2 , . . . , xd )
(7.1.1)
s
d x1 · · · d xd
for various particular functions f . Two instances were studied in detail.
(1) First, the two-dimensional case where f is given by a positive definite
binary quadratic form Q(x, y) := ax 2 + bx y + cy 2 was analyzed in
[5, Theorem 1].
Namely, for any positive definite form Q, the quantity σ (s) :=
lim N →∞ σ N (s) exists in the strip 0 < Re s < 1 and coincides therein
with the analytic continuation of α(s).
Thus, the integral β N (s) plays no role in the final answer. This is very
tidy for two-dimensional lattices.
7.2 Commentary: Sums related to the Poisson equation
237
(2) The authors of [5] then investigated the three-dimensional case for the simple cubic lattice, namely Q(x, y, z) := x 2 + y 2 + z 2 and came to a similar
conclusion as before, for the strip 12 < Re s < 32 . However, at s = 12 , σ (s)
is discontinuous and
σ 12 = α 12 + 16 π.
(7.1.2)
The physicist’s method of finding lim N →∞ σ N ( 12 ) always alights on α( 12 );
why this is so is not fully understood but some heuristic explanations–
regarding analytic continuations – are to be found in [6].
We leave an open question:
Can a similar analysis be done for the four-dimensional simple cubic lattice?
Presumably, there is a strip for which σ (s) = α(s) but presumably s = 12
will not lie within this strip?
In four dimensions the closed form is known for α in the simple cubic case:
α(s) = 8 1 − 22−2s ζ (s)ζ (s − 1);
see [8, (9.2.5)].
7.2 Commentary: Sums related to the Poisson equation
In a recent treatment of ‘natural’ Madelung constants [11], it is pointed out that
the Poisson equation for an n-dimensional point-charge source,
∇2
n (r)
= −δ(r),
(7.2.1)
gives rise to an electrostatic potential – we call it the bare-charge potential – of
the form
( n2 − 1) 1
Cn
=: n−2
4π n/2 r n−2
r
1
log r =: C2 log r,
2 (r) = −
2π
n (r)
=
if n = 2,
(7.2.2)
(7.2.3)
where r := |r|. Since this Poisson solution generally behaves as r 2−n , [11] defines
a ‘natural’ Madelung constant Nn as (here, m := |m|):
Nn := Cn
(−1)1·m
if n = 2,
m n−2
n
m∈Z
N2 := C2
m ∈ Zn
(−1)1·m log m,
(7.2.4)
238
Electron sums
where, in cases such as this log sum, one must infer an analytic continuation [11]
as the literal sum is quite non-convergent. This Nn coincides with the classical
Madelung constant
(−1)1·m
Mn :=
m
n
(7.2.5)
m∈Z
1
only for n = 3 dimensions, in which case N3 = 4π
M3 . In all other dimensions
there is no obvious M, N relation.
A method for gleaning information about Nn is to contemplate the Poisson
equation with a crystal charge source, modelled on NaCl (salt) in the sense of
alternating lattice charges:
(−1)1·m δ(m − r).
(7.2.6)
∇ 2 φn (r) = −
m ∈ Zn
Accordingly – on the basis of the Poisson equation (7.2.1) – solutions φn can be
written in terms of the respective bare-charge functions n as
φn (r) =
(−1)1·m n (r − m).
(7.2.7)
m ∈ Zn
7.2.1 Madelung variants
We have defined the classical Madelung constants (7.2.5) and the ‘natural’
Madelung constants (7.2.4). Following [11] we define a Madelung potential, now
depending on a complex s and spatial point r ∈ Zn :
(−1)1·p
,
|p − r|s
n
Mn (s, r) :=
(7.2.8)
p∈Z
We can write limit formulae for our Madelung variants, first the classical
Madelung constant,
1
(7.2.9)
Mn := lim Mn (1, r) −
r→0
r
(−1)1·p
=
,
p
n
(7.2.10)
p∈Z
and then the ‘natural’ Madelung constant,
Nn := lim [φ(r) −
r→0
(r )]
(7.2.11)
(−1)1·p
= Cn
.
p n−2
n
p∈Z
(7.2.12)
7.2 Commentary: Sums related to the Poisson equation
239
For small even n, this last sum is evaluable. For example, from [8, (9.2.5)] we have
(−1)1·p
= (1 − 22−s )(1 − 21−s )ζ (s)ζ (s − 1),
p 2s
4
(7.2.13)
p∈Z
which with s = 1 yields
N4 = −
1
log 2.
π2
Similarly, from [8, Exercise 4b, p. 292] we have
(−1)1·p
= −16(1 − 24−s )ζ (s)ζ (s − 3),
2s
p
8
(7.2.14)
p∈Z
which with s = 3 determines that
N8 = −
4
ζ (3).
π4
Generally, via the Mellin transform Ms θ42n (q) (see below), values of N2n for
small n are similarly susceptible. For instance, if G denotes Catalan’s constant,
we have
2G
1
− 3,
N6 = −
24π
π
as in [11]. The more complex value N2 is presented in (7.2.21), below.
7.2.2 Relation between crystal solutions φn and Madelung potentials
From (7.2.2), (7.2.7), and (7.2.8) we have the general relation, for dimension
n = 2,
φn (r) = Cn Mn (n − 2, r).
(7.2.15)
Note that, for the case n = 3, the solution φ3 coincides with the classical
Madelung potential M3 (1, r) in the sense that
φ3 (r) =
1
M3 (1, r),
4π
because C3 = 1/(4π ). Likewise, the ‘natural’ and classical Madelung constants
are related by 4π N3 = M3 . The whole idea of introducing ‘natural’ Madelung
constants Nn is that this coincidence of radial powers for φ and M potentials
holds only in three dimensions. For example, in n = 5 dimensions, the summands
for φ5 and M5 (1, ·) involve radial powers 1/r 3 , as in
φ5 (r) =
1
M5 (3, r).
8π 2
240
Electron sums
In [11] it is argued that a solution to (7.2.6) is
n
1 k=1 cos π m k rk
φn (r) = 2
,
π
m2
n
(7.2.16)
m∈O
where O denotes the odd integers (including negative odds). These φn give the
potential within the appropriate n-dimensional crystal, in that φn vanishes on the
surface of the cube [− 12 , 12 ]n , as is required via symmetry within an NaCl-type
crystal of any dimension. To render this representation more explicit and efficient,
we could write equivalently
φn (r) =
2n
π2
m 1 ,...,m n
cos π m 1r1 · · · cos π m n rn
.
m 21 + · · · + m 2n
> 0, odd
It is also useful that – owing to the symmetry arising because the summation
indices are odd – we can replace in a cavalier way the cosine product in (7.2.16)
with a simple exponential:
φn (r) =
1 eiπ m·r
.
π2
m2
n
(7.2.17)
m∈O
7.2.3 Fast series for φn
From previous work [11] we know a computational series
1 sinh[π R(1/2 − |r1 |] n−1
k=1 cos π Rk rk+1
,
φn (r) =
2π
R
cosh
(π
R/2)
n−1
(7.2.18)
R∈O
suitable for any nonzero vector r ∈ [− 12 , 12 ]n . The work [11] also gives an
improvement in the case of n = 2 dimensions, namely the following form (see
Fig. 7.1) for which the logarithmic singularity at the origin has been siphoned off:
φ2 (x, y) =
cosh π x + cos π y
2 cosh π mx cos π my
1
log
−
.
4π
cosh π x − cos π y
π
m(1 + eπ m )
+
m∈O
(7.2.19)
These series, (7.2.18) and (7.2.19) are valid, respectively, for r1 , x ∈ [−1, 1]. For
clarification, we give here the fast series for n = 3 dimensions:
π
2 + q 2 (1 − 2|x|) cos π py cos πqz
p
sinh
2
2
φ3 (x, y, z) =
.
π
p 2 + q 2 cosh π p 2 + q 2
p,q > 0, odd
2
(7.2.20)
7.2 Commentary: Sums related to the Poisson equation
241
7.2.4 Closed form for the ‘natural’ Madelung constant N2
The natural Madelung constant for n = 2 dimensions has also been found, on the
basis of (7.2.19) (see [11]), to take the value
N2 =
4 3 ( 34 )
1
log
.
4π
π3
(7.2.21)
We remind ourselves that this closed form was achieved by contemplating the limiting process r → 0 and hence by Coulomb-singularity removal. The derivation
of the above N2 form depends on the relation
4 3
3
4
π3
√
=
K3/2
2
√1
2
π 3/4
.
We also record the following numerically effective Mellin transform for
n > 2:
∞
%
&
1
1 − θ4n e−π x x n/2−2 d x < 0,
(7.2.22)
Nn = −
4π 0
where the integral is positive since 0 < θ4 (q) < 1 for 0 ≤ q ≤ 1. From this the
large-n behaviour of Nn may be estimated as
Nn −
( 12 n − 1)
π n/2
n
n(n − 1)
−
+
·
·
·
,
2
2n/2
(7.2.23)
on making the approximations
θ4 (q) = 1 − 2q + O q 4
and
1 − x n = −n(x − 1) +
&
%
n(n − 1)
(x − 1)2 + O (x − 1)3
2
and then integrating term by term. For instance, from (7.2.22) we compute
N100 = −8.6175767047403040779 . . . × 1037
while the asymptotic (7.2.23) gives
N100 −8.6175767047403038 . . . × 1037 .
242
Electron sums
7.2.5 Closed forms for the φ2 -potential
Theorem 7.1 It can be proved that
2
1
log 1 + √ ,
=
8π
3
√
1
1 1
φ2 ( 4 , 4 ) =
log(1 + 2),
4π
√
1
log(3 + 2 3).
φ2 ( 13 , 0) =
8π
φ2 ( 13 , 13 )
(7.2.24)
(7.2.25)
(7.2.26)
Proof Consider, for s > 0,
V2 (x, y; s) :=
∞
[cos π(2m + 1)x][cos π(2n + 1)y]
.
[(2m + 1)2 + (2n + 1)2 ]s
m,n=−∞
(7.2.27)
This V2 -function is related to φ2 by V2 (x, y; 1) = π 2 φ2 (x, y). Treating it as a
general lattice sum [8], we derive (with some difficulty)
V2 13 , 13 ; s =2−1−s −(1 − 2−s )(1 − 32−2s )L 1 (s)L −4 (s)
+ 3(1 + 2−s )L −3 (s)L 12 (s) .
(7.2.28)
The L-functions in (7.2.28) are various Dirichlet series; L 1 is the Riemann
ζ -function. Note that 1 − 32−2s factors as (1 + 31−s )(1 − 31−s ), that lims→1
(1 − 31−s )L 1 (s) = log 3, and that
√
√
3π
π
1
L −4 (1) = ,
L −3 (1) =
,
L 12 (1) = √ log(2 + 3). (7.2.29)
4
9
3
After gathering everything together we have
φ2 ( 13 , 13 )
√ 3+2 3
1
1
1 1
log
= 2 V2 ( 3 , 3 , 1) =
,
8π
3
π
which is (7.2.24).
We find more easily that
V2 ( 14 , 14 ; s) = 2
∞
m,n=−∞
(−1)m+n
= 21−s L −8 (s)L 8 (s)
[(4m − 1)2 + (4n − 1)2 ]s
(7.2.30)
is a familiar lattice sum [8]. So, with
√
π
1
and
L 8 (1) = √ log(1 + 2),
L −8 (1) = √
2 2
2
we derive
√
1
φ2 ( 14 , 14 ) = 4π
log(1 + 2),
(7.2.31)
which is (7.2.25). Likewise
V2 13 , 0; s = 2−1−s (1 − 2−s )(1 − 32−2s )L 1 (s)L −4 (s)
+ 3(1 + 2−s )L −3 (s)L 12 (s) ,
(7.2.32)
7.2 Commentary: Sums related to the Poisson equation
243
which yields
√ π
√
1
φ2 0, 13 =
log 3(2 + 3)2 = log(3 + 2 3),
16π
8
which is (7.2.26).
Remark: The V2 lattice sum can be given as a fast series:
cos π mx cos π ny
(7.2.33)
V2 (x, y; s) :=
(m 2 + n 2 )s
m,n ∈ O
23/2−s π s |u + x| s−1/2
=
(−1)u
K 1/2−s (π n|u + x|) cos π ny,
(s)
n
+
n∈O
u ∈Z
where K ν is a standard modified Bessel function. For s = 1 this series collapses
further into π 2 times our series (7.2.19) for the Poisson potential φ2 .
Using the integer relation method PSLQ [7] to hunt for results of the form
exp π φ2 (x, y)
?
= α,
(7.2.34)
for α algebraic, we may obtain and further simplify many results such as the
following.
Conjecture 7.1 Empirically,
α + 1/α √
1
?
log α where
= 2 + 1,
φ2 14 , 0 =
4π
2
'
√
√
1
?
log 3 + 2 5 + 2 5 + 2 5 ,
φ2 15 , 15 =
8π
√
γ + 1/γ
1
?
log γ where
= 3 + 1,
φ2 16 , 16 =
4π
2
√
τ − 1/τ
1
?
1 1
log τ where
= (2 3 − 3)1/4 ,
φ2 3 , 6 =
4π
2
√ 1+ 2− 2
1
?
φ2 18 , 18 =
log
,
√
4
4π
2−1
'
√
√
μ + 1/μ
1
?
1 1
log μ where
= 2 + 5 + 5 + 2 5;
, 10 =
φ2 10
4π
2
?
the notation = indicates that we have only experimental (i.e., extreme-precision
numerical) evidence of an equality.
Such hunts are made entirely practicable by (7.2.19). Note that for general x
and y we have φ2 (y, x) = φ2 (x, y) = −φ2 (x, 1 − y), so we can restrict searches
to 12 > x ≥ y > 0, as illustrated in Fig. 7.1. Indeed computations by Glasser
and Crandall given in [11] were precipitated from such experiments and led to the
following result.
244
Electron sums
0.5
0.0
–0.5
0.6
0.4
0.2
0.0
–0.5
0.0
0.5
Figure 7.1 High-precision plot of the Monge surface z = φ2 (x, y), via the fast series
(7.2.19), showing the logarithmic singularity at the origin (the plot is adapted from [11]).
In this plot, x, y range over the 2-cube [− 12 , 12 ]2 ; from symmetry one need know the φ2
surface only over the octant 12 > x ≥ y > 0. We are able to establish closed forms for the
heights on this surface above certain rational √
pairs (x, y). As just one example,
1 log(1 + 2) ≈ 0.0701.
φ2 ( 14 , 14 ) = 4π
Theorem 7.2 For z :=
π
(y + i x),
2
1 − λ(z)/√2 θ2 (z, q)θ4 (z, q) 1
1
=
φ2 (x, y) =
log log √ , (7.2.35)
2π
θ1 (z, q)θ3 (z, q)
4π
1 − 1/[λ(z) 2] where
λ(z) =
θ42 (z, e−π )
θ32 (z, e−π )
=
∞
(1 − 2 cos 2z q 2n−1 + q 4n−2 )2
,
(1 + 2 cos 2z q 2n−1 + q 4n−2 )2
(7.2.36)
n=1
with q := e−π .
Hence, the presumption that, for rational x, y, (7.2.34) is always algebraic is
equivalent to the provable conjecture that, for all z = π2 (y +i x) with x, y rational,
λ(z) in (7.2.36) is algebraic, and this is proved in [1, Theorem 10]. How to provide
computationally assisted proofs of results such as those of Conjecture 7.1 is also
discussed in [1].
References
245
7.2.6 Closed forms for n = 3, 4 dimensions
The only nontrivial closed-form evaluation of φ3 of which we are aware is that of
Forrester and Glasser [8, 13]. Namely,
4π φ3 (1/6) = M3 (1, 1/6) =
m ∈ Z3
√
(−1)1·m
= 3.
|m − 1/6|
References
[1] D. H. Bailey, J. M. Borwein, R. E. Crandall, and I. J. Zucker. Lattice sums arising
from the Poisson equation. J. Phys. A, 46:115201–115232, 2012.
[2] A. Baldereschi, G. Senatore, and I. Oriani. Madelung energy of the Wigner
crystal on lattices with non-equivalent sites. Solid State Commum., 81:21–22,
1992.
[3] L. Bonsall and A. A. Maradudin. Some static and dynamical properties of a twodimensional Wigner crystal. Phys. Rev. B, 15:1959–1973, 1977.
[4] M. Born and R. D. Misra. On the stability of crystal lattices. IV. Proc. Camb. Phil.
Soc., 36:466–478, 1940.
[5] D. Borwein, J. M. Borwein, and R. Shail. Analysis of certain lattice sums. J. Math.
Anal. Appl., 143:126–137, 1989.
[6] D. Borwein, J. M. Borwein, R. Shail, and I. J. Zucker. Energy of static electron
lattices. J. Phys. A: Math. Gen., 21:1519–1531, 1988.
[7] J. M. Borwein and D. H. Bailey. Mathematics by Experiment: Plausible Reasoning
in the 21st Century, 2nd edition. A. K. Peters, 2008.
[8] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number
Theory and Computational Complexity. Wiley, New York, 1987.
[9] E. Cockayne. Comment on ‘stability of the Wigner electron crystal on the perovskite
lattice’. J. Phys. Condens. Matter, 3:8757, 1991.
[10] R. A. Coldwell-Horsfall and A. A. Maradudin. Zero point energy of an electron
lattice. J. Math. Phys., 1:395–404, 1960.
[11] R. E. Crandall. The Poisson equation and ‘natural’ Madelung constants. 2012.
Preprint.
[12] L. L. Foldy. Electrostatic stability of Wigner and Wigner–Dyson lattices. Phys. Rev.
B, 17:4889–4894, 1978.
[13] P. J. Forrester and M. L. Glasser. Some new lattice sums including an exact result
for the electrostatic potential within the NaCl lattice. J. Phys. A, 15:911–914,
1982.
[14] K. Fuchs. A quantum mechanical investigation of the cohesive forces of metallic
copper. Proc. Roy. Soc. London A, 151:585–602, 1935.
[15] F. Hund. Versuch einer Ableitung der Gittertypen aus der Vorstellung des isotropen
polarisierbaren Ions. Z. Phys., 34:833–857, 1925.
[16] F. Hund. Vergleich der elektrostatischen Energien einiger Ionengitter. Z. Phys.,
94:11–21, 1935.
246
Electron sums
[17] E. P. Wigner. On the interaction of electrons in metals. Phys. Rev., 46:1002–1011,
1934.
[18] I. J. Zucker. Stability of the Wigner electron crystal on the perovskite lattice. J. Phys.
Condens. Matter, 3:2595–2596, 1991.
[19] I. J. Zucker and M. M. Robertson. Systematic approach to the evaluation of
2
2 −s
(m,n=0,0) (am + bmn + cn ) . J. Phys. A, 9:1215–1225, 1976.
8
Madelung sums in higher dimensions
Here we make a general study of the convergence properties of lattice sums,
involving potentials, of the form occurring in mathematical chemistry and
physics. Many specific examples are studied in detail. The prototype is the
Madelung constant for NaCl, presuming that one appropriately interprets the
summation process.
8.1 Introduction
Lattice sums of the form arising in crystalline structures – and defined precisely
in the next section – have been subject to intensive study. A very good overview is
available in [7] and related research may be followed up in [4–6]. These sums are
highly conditional in their convergence and the subject of how best to interpret
their convergence is discussed in [3–6] and the references therein. The prototype
is the Madelung constant for NaCl:
∞
−∞
(−1)n+m+ p
= −1.74756459 . . . ,
n 2 + m 2 + p2
presuming that one sums over expanding cubes but not spheres [4].
Since the analytic or numerical evaluations of such sums usually proceed by
transform (and renormalization) methods, these issues are often obscured, especially in the physical science literature. As we shall illustrate in this paper, while
some general theorems are available the precise study of convergence is a delicate
and varied subject. Some of our results are unsurprising, others far from intuitive.
248
Madelung sums in higher dimensions
8.2 Preliminaries and notation
We shall suppose throughout that
Q(x1 , . . . , xk ) :=
k k
αi j xi x j ∈ R[x1 , . . . , xk ]
i=1 j=1
is a positive definite quadratic form with αi j = α ji . For a bounded set C in Rk
and a positive real number ν we understand νC to be the set (u 1 , . . . , u k ) ∈ Rk
such that (u 1 /ν, . . . , u k /ν) ∈ C and write
Cν := νC ∩ (Zk \ (0, . . . , 0)).
We shall chiefly be interested in C ⊂ Rk , where (0, . . . ,0) lies in the interior of C,
so that
lim νC = Rk .
ν→∞
We define the corresponding lattice sum
Aν (s) = Aν (C, Q, s) :=
(x1 ,...,xk )∈Cν
(−1)x1 +···+xk
Q(x1 , . . . , xk )s
and write
A(s) = A(C, Q, s) := lim Aν (C, Q, s)
ν→∞
whenever this limit exists. For the most part we shall suppress explicit reference
to parameters such as C and Q and simply write A(s) (except in Sections 8.4.2
and 8.7, where we use A(C, Q, s) to emphasize the dependence upon the region
C). Throughout, we avoid summing over the pole at zero. Also we often write
σ for Re s. (The literature is split as to whether to write A(s) or A(2s); the latter form moves the physically meaningful value from 12 to 1.) At s := 12 our
sums are evaluating weighted and signed potentials at the origin over points in the
underlying lattice.
While we have stated our results with reference to integer lattice points, we
x ) by
can readily generate analogues for an arbitrary lattice AZk on replacing Q(
Q(A x). Notice that a convex body is mapped to a convex body by the matrix A−1 .
Our key result is to show that A(s) exists and is analytic at least down to
Re s > 12 (k − 1) for all reasonably shaped regions C (and hence that the limit
is independent of the shape C chosen in that range). In fact, as the next section shows, the same is true if we replace the factor (−1)x1 +···+xk by a function
q(x1 , . . . , xk ) exhibiting a similar degree of cancellation when summed over one
of the xi .
In Section 8.4 we examine in detail the question of convergence for Re s ≤
1
(k
− 1) when C is an (appropriate) ellipse in Rk , an arbitrary polygon in R2 or
2
R3 with rational vertices, or a k-dimensional rectangle (showing that in the latter
8.3 A convergence theorem for general regions
249
case convergence actually holds for all Re s > 0). In Section 8.5 we give very
explicit formulae when Q(x, y) := x 2 + P y 2 for certain P (particularly for P = 3
or 7) and C is the corresponding ellipse x 2 + P y 2 ≤ 1. Several other examples
are detailed in Section 8.6. Finally, in Section 8.7, when Q(x, y) := x 2 + y 2 we
demonstrate directly the existence and equivalence of the limits at s = 1 (the most
analytically pliable value) for C a circle, rectangle, or diamond. Since many of the
proofs are lengthy and technical we have chosen to postpone the majority of them
until Section 8.8.
8.3 A convergence theorem for general regions
Let C be a bounded set in Rk containing (0, . . . , 0) in its interior and let
Q(x1 , . . . , xk ) be a positive definite quadratic form in R[x1 , . . . , xk ]. Let q :
Zk → R, and let
q(x1 , . . . , xk )
.
Aν (s) = Aν (C, Q, q, s) :=
Q(x1 , . . . , xk )s
(x1 ,...,xk )∈Cν
We may now state our basic result.
Theorem 8.1
Let χν be the characteristic function of Cν , and suppose that
wν ( j1 , . . . , jk−1 , m) :=
m
q( j1 , . . . , jk−1 , l)χν ( j1 , . . . , jk−1 , l)
l=−∞
is uniformly bounded for all integers j1 , . . . , jk−1 , m and all positive ν. Then
A(s) = A(C, Q, q, s) := lim Aν (s)
ν→∞
exists and is analytic in the region σ := Re s > 12 (k − 1).
In general, this is the most that we can say, as can be seen by taking C to be (for
example) the l1 ball in Rk :
Theorem 8.2 Suppose that q(x1 , . . . , xk ) := (−1)x1 +···+xk and that C is the
k-dimensional diamond a1 |x1 | + · · · + ak |xk | ≤ c, where a1 , . . . , ak , c ∈ N with
d = gcd(a1 , . . . , ak ) and all the ai /d odd; then Aν ( 12 (k − 1)) does not tend to a
limit as ν → ∞.
Theorem 8.1 shows that in R2 the limit is well defined for σ > 12 , and in R3
for σ > 1, for any sensible region and any reasonable q. We make this precise in
the corollary below. We say that a region C in Rk is convex in the ith variable if
whenever the points (x1 , . . . , xi , . . . , xk ) and (x1 , . . . , xi , . . . , xk ) are in C then
so also is the segment joining them.
250
Madelung sums in higher dimensions
Corollary 8.1 Suppose that q(x1 , . . . , xi , . . . , xk ) is bounded over Zk and is
periodic with period M in one variable, xi say, with
M
q(x1 , . . . , xk ) = 0.
xi =1
Suppose further that C is bounded, contains (0, . . . , 0) in its interior, and is
convex in the ith variable. Then the conclusions of Theorem 8.1 hold.
Indeed, many highly non-convex regions still satisfy Theorem 8.1. We will refer
to a vertically convex region in Rk as being one in which the final coordinate
exhibits convexity.
We now focus on sums over specific regions, showing that in some cases
convergence can continue well below σ = 12 (k − 1).
8.4 Specific regions
8.4.1 Lattice sums over sympathetic ellipses
k k
Given a positive definite quadratic form Q(x1 , . . . xk ) = i=1
j=1 αi j x i x j ∈
Z[x1 , . . . , xk ], with αi j = α ji , and function q : Zk → R we define the arithmetic
function
q(
x ).
r (n, Q, q) :=
Q(
x )=n
x∈Zk
In particular when our quadratic form has integer coefficients and we sum over
the lattice points in appropriate ellipses Q(x1 , . . . , xk ) ≤ ν, we can replace the
k-dimensional lattice sum by a Dirichlet series
q(
r (n, Q, q)
x)
=
Aν (s) = Aν (Q, q, s) :=
s
Q(
x)
ns
Q(
x )≤ν
x∈Zk \0
1≤n≤ν
and decide when the limit
A(s) = A(Q, q, s) := lim
ν→∞
exists by examining the sums
S0 (x) = S0 (Q, q, x) :=
r (n, Q, q)
ns
1≤n≤ν
r (n, Q, q).
(8.4.1)
0≤n≤x
We recall the formula (see Hardy and Riesz [8, Theorem 7]) for the abscissa of
convergence σ0 > 0 of such a Dirichlet series,
σ0 = lim sup
x→∞
log |S0 (x)|
.
log x
(8.4.2)
8.4 Specific regions
251
That is (see [8, Theorem 1]), A(s) will exist for all Re s > σ0 and fail to exist for
all Re s < σ0 .
We now show that (at least for periodic q with suitable cancellation when
summed) convergence over these ellipses always extends below σ = 12 (k − 1).
Theorem 8.3
xi . If
Suppose that q(x1 , . . . , xk ) is periodic with period M in each
M
···
r1 =1
M
q(r1 , . . . , rk ) = 0
rk =1
then the abscissa of convergence σ0 satisfies
⎧
⎨ 23/73
0 < σ0 ≤
25/34
⎩
k/2 − 1
if k = 2,
if k = 3,
if k ≥ 4.
If
M
···
r1 =1
M
q(r1 , . . . , rk ) = 0
rk =1
then the abscissa of convergence σ0 = k/2.
The theorem follows easily from work done in the past by Landau and Walfisz
(for k ≥ 4) and the more recent bounds of Krätzel and Nowak (for k = 3) [11] and
Huxley [9] (for k = 2), on the error in approximating the number of lattice points
in an ellipsoid by its volume. When k ≥ 4 the bounds in Theorem 8.3 cannot in
general be improved.
Theorem 8.4 If q(x1 , . . . , xk ) := (−1)x1 +···+xk and Q(x1 , . . . , xk ) := a1 x12 +
· · · + ak xk2 , where the ai are all odd positive integers, then for all k ≥ 2 the limit
A(s) does not exist for any Re s ≤ 12 k − 1.
When k = 2 or 3 one expects the correct upper bounds to be 14 and 12 respectively. From Theorem 8.4, the bound 12 would certainly be sharp when k = 3.
In fact we will show that for very general Q and q we have the lower bound
σ0 ≥ 14 (k − 1) so that when k = 2 or 3 usually we do indeed have the lower
bounds 14 and 12 :
Theorem 8.5
M and
Suppose that q(x1 , . . . , xk ) is periodic in all the xi , with period
M
r1 =1
···
M
rk =1
q(r1 , . . . , rk ) = 0.
252
Madelung sums in higher dimensions
If r (n, Q, q) (and hence Aν (s)) is not identically 0 then
σ0 ≥ 14 (k − 1).
Notice that there certainly will be cases with Aν (s) identically zero (with
therefore no lower bound on σ0 ); indeed for any M = 2 we can always construct non-trivial periodic q(
x ) with q(−
x ) = −q(
x ) and hence, by symmetry,
r (n, Q, q) zero for all n and any Q.
For the proof of Theorem 8.5 we will use a technique of Landau to show the
existence of a constant c0 = c0 (q, Q) > 0 such that
|S0 (x)| > c0 x (k−1)/4
for infinitely many integers x. The method requires some additional notation.
k k
Given a positive definite quadratic form Q(x1 , . . . , xk ) =
i=1
j=1 αi j x i x j
in Z[x] with αi j = α ji we let D denote the determinant
α11 . . . α1k .. (8.4.3)
D := ...
. α
k1 . . . αkk
and define the positive definite adjoint quadratic form Q ∗ (x1 , . . . , xk ) in Z[x] by
Q ∗ (x1 , . . . , xk ) :=
k k
∂D
xi x j .
∂αi j
i=1 j=1
Q ∗∗ (
x)
Notice that
= Q(
x ) and that when k = 2 we have Q ∗ (x, y) = Q(−y, x).
We suppose that q(x1 , . . . , xk ) is periodic in each xi with period M, define the
periodic weight function
M
M
2π
(r1 x1 + · · · + rk xk ) ,
x ) :=
···
q(r1 , . . . , rk ) cos
λq (
M
r1 =1
rk =1
and set
r ∗ (n) = r ∗ (n, Q, q) :=
λq (
x ).
x∈Zk
Q ∗ (
x )=n
For the q(
x ) of interest the involved-looking expression for r ∗ (n) often simplifies.
For example, when
q(x1 , . . . , xk ) := (−1)x1 +···+xs ,
for some 1 ≤ s ≤ k, we have
r ∗ (n) = 2k
Q ∗ (x1 ,...,xk )=n
x1 ,...,xs odd
xs+1 ,...,xk even
1.
8.4 Specific regions
253
We remark that r (n, Q, q) is identically zero if and only if r ∗ (n) is identically
zero. As with the classical circle problem, our proof relies on the ability to write
the sum S0 (x) in terms of Bessel functions Jν (x):
√
( ∞
2π nx
( D M)(k−2)/2 k/4 r ∗ (n)
∗
x
,
r (n, Q, q) =
Jk/2
k/4
M
M D
n
0≤n≤x
n=1
∗
where
denotes that if x is an integer the last term in the sum receives only
half weighting: it is 12 r (x, Q, q). In fact we shall actually use integrated forms of
this that are more surely convergent.
As one consequence of the proof, defining
Sν (x) :=
1 (x − n)ν r (n, Q, q)
ρ!
0≤n≤x
and setting
Bν(ρ) (s) := Aν (s) −
ρ−1
i=0
(s + i)
(s)
Si (ν) q(0)
−
ν s+i
i!
+
(s + ρ) q(0)
,
(s)
ρ!
it will be apparent that for any positive integer ρ > 12 (k − 1) we can write
Bν(ρ) (s) = A(ρ) (s) + O ν −(ρ−(k−1)/2+2σ )/2 ,
where Aρ (s) is analytic in the larger region
,
σ > − 12 ρ − 12 (k − 1) + ε, |s| < K ,
for any fixed K and ε > 0, and possesses the representation
√
D M k−2s−1
2k/2+ρ π k/2−1 (s + ρ + 1)
ρ
A (s) =
M
(s)
2π
(
∞
∗
2π n
r (n)
,s
Fρ
k/2−s
M D
n
(8.4.4)
n=1
with
Fρ (z, s) := z 2s−k
∞
v k/2−1−ρ−2s Jk/2+ρ (v) dv.
z
When k = 3 we note some similarity to the relation of Buhler and Crandall
[6, (1.5)].
Finally, for k ≥ 4, the work of Novák enables us to extend the optimal bound
σ0 = k/2 − 1 of Theorem 8.4 to a broader (if less easily described) class of q and
254
Madelung sums in higher dimensions
Q. We shall say that we are in the non-singular case if, as before, q(x1 , . . . , xk )
is periodic in all xi , with period M and
M
r1 =1
···
M
q(r1 , . . . , rk ) = 0,
rk =1
and, moreover there exist integers h and l > 0 with (h, l) = 1 and
lM
r1 =1
···
lM
rk =1
2πi h
Q(r1 , . . . , rk ) = 0.
q(r1 , . . . , rk ) exp −
l
Note that any such l necessarily has (l, M) = 1.
Theorem 8.6
have
For non-singular pairs of q(x1 , . . . , xk ) and Q(x1 , . . . , xk ) we
σ0 ≥
k
− 1.
2
We shall show in the following corollary that any q(x1 , . . . , xk ) of the form
q(x1 , . . . , xk ) := (−1)x1 +···+xs ,
1 ≤ s ≤ k,
is non-singular for all Q(x1 , . . . , xk ), so that we certainly recover Theorem 8.4 by
this approach (of course the proof of Theorem 8.6 will be much less elementary
than that of Theorem 8.4).
Corollary 8.2
If q(x1 , . . . , xk ) is of the form
2πi
(a1 x1 + · · · + ak xk )
q(x1 , . . . , xk ) := exp
M
for some integers ai then the case is non-singular and
σ0 ≥
k
−1
2
for all positive definite quadratic forms Q(x1 , . . . , xk ) in Z[x1 , . . . , xk ].
When q(x1 , . . . , xk ) takes the special form exp 2πi
M (a1 x 1 + · · · + ak x k ) if
xi ≡ bi (mod Mi ) for some integers ai , bi and Mi (with Mi |M) and is zero otherwise, Walfisz [21] showed (see Novák [17]) that in the singular case the upper
1
for k > 4. Thus this division into singular
bound can be lowered to σ0 ≤ 14 k − 10
and non-singular cases (although not immediately digestible) is probably the correct characterization as regards the abscissa, and Theorem 8.5 conceivably gives
the best general lower bound.
8.4 Specific regions
255
8.4.2 Lattice sums over polygons in R2 and R3
In this subsection we restrict ourselves to the usual weight function
q(x1 , . . . , xk ) := (−1)x1 +···+xk . We write A(C, Q, s) for the corresponding lattice sum, rather than merely A(s), to emphasize the dependence here upon the
region C (the Q-dependence is of use in the proof of Theorem 8.8 below). For
polygons in R2 with rational vertices, we show that either convergence occurs for
all Re s > 0 or else convergence fails at s = 12 . Moreover we give an explicit
and somewhat surprising diophantine criterion for deciding this, on the basis of
the parity of the numerators and denominators of the slopes of the lines making
up the perimeters of the polygons.
Theorem 8.7 Suppose that P in R2 is a closed polygon with rational vertices whose sides lie on the lines ai x − bi y = ci , where ai , bi , ci ∈ Z with
gcd(ai , bi , ci ) = 1 and di = gcd(ai , bi ).
(i) If ai /di and bi /di are of opposite parity for all i then A(P, Q, s) exists and
is analytic for all Re s > 0.
(ii) If ai /di and bi /di are both odd for at least one i then A(P, Q, s) exists
for Re s > 12 but does not exist for any real s ≤ 12 . If in addition P is star-shaped around (0, 0) then, even restricting n to an integer,
lim An (P, Q, s) does not exist for any real s ≤ 12 .
n→∞
We extract two simple cases for further advertisement in the next corollary:
Corollary 8.3
(i) If R is a rectangle with rational vertices and sides parallel to the axes then
A(R, Q, s) exists for Re s > 0.
(ii) If D is the diamond a|x|+b|y| ≤ c, where a, b, c ∈ N with gcd(a, b, c) = 1
and d = gcd(a, b), then A(D, Q, s) exists for Re s > 0 if a/d and b/d are
of opposite parity but fails to exist for any real s ≤ 12 if a/d and b/d are
both odd.
We note that this last corollary allows us to observe that in the Hausdorff metric,
or any other reasonable metric, the convex bodies for which convergence works
for all σ > 0 are dense in the convex bodies in the unit ball, as are those for which
convergence is lost for s = 12 .
Theorem 8.6 follows from a more precise version that (for Re s > 0) reduces
the problem of convergence over the polygon to the question of convergence
solely along the boundary. If we define a variant of the characteristic function
256
Madelung sums in higher dimensions
χC∗ , where points on the boundary of C receive weight 12 , and a corresponding
analogue of Aν (P, Q, s),
A∗ν (C, Q, s) :=
(−1)x+y
(x,y)∈Cν
∗ (x, y)
χνC
,
Q(x, y)s
then the following is true.
Proposition 8.1 Let Q(x, y) = αx 2 +βx y+γ y 2 be a positive definite quadratic
form and let P be a closed polygon in R2 with rational vertices whose sides lie
on the lines ai x − bi y = ci , where ai , bi , ci ∈ Z with gcd(ai , bi , ci ) = 1. If N is
a multiple of all the gcd(ai , bi ) then
A∗N (P, Q, s) = F(P, Q, s) + O P,Q (N −2σ ),
where F(P, Q, s) is analytic in the whole half-plane σ := Re s > 0.
Similarly, in three dimensions we can show that convergence either fails at
s = 1 or continues down to s = 12 .
Theorem 8.8 Suppose that P is a three-dimensional polygon, containing 0 in its
whose faces lie on the planes ai x + bi y + ci z =
interior and star-shaped about 0,
ei where ai , bi , ci and ei are integers with di = gcd(ai , bi , ci ).
(i) If every face has at least one of ai /di , bi /di , ci /di even, then A(P, Q, s)
exists for all Re s > 12 .
(ii) If there is at least one face with ai /di , bi /di , ci /di all odd, then A(P, Q, s)
exists for all Re s > 1 but fails to exist for any real s ≤ 1.
We presume that these two- and three-dimensional theorems can be generalized to more dimensions. We have already seen in Theorem 8.2 that the result
on diamonds extends naturally to higher dimensions. In the next subsection we
show that the behaviour over squares can be similarly recaptured in arbitrary
dimensions.
8.4.3 Sums over rectangles
When we sum over k-dimensional rectangles we are able to show that, in general,
convergence holds for all Re s > 0. More precisely, given an m
= (m 1 , . . . , m k )
in Nk we define the lattice sum over the corresponding rectangle:
Am (s) :=
m1
n 1 =−m 1
···
mk
n k =−m k
q(n 1 , . . . , n k )
Q(n 1 , . . . , n k )s
8.5 Some analytic continuations
257
(where as usual the pole (n 1 , . . . , n k ) = (0, . . . , 0) is omitted) and set
A(s) :=
lim
min m i →∞
Am (s)
whenever that limit exists.
Theorem 8.9
If the sums
l1
n 1 =− j1
···
lk
q(n 1 , . . . , n k )
n k =− jk
are uniformly bounded for all integers j1 , . . . , jk , l1 , . . . , lk then the limit A(s)
exists and is analytic for all σ > 0.
8.4.4 l p -balls: an open question
It is natural to make an examination of l p -sums for p ∈ N, p > 2, that is, when
k = 2 for example, C := {(x, y) : |x| p + |y| p ≤ 1}. We are able to state (see
[10]) asymptotically sharp expressions for the number of lattice points in these
regions:
1=
|n| p +|m| p ≤x
2 2 ( p −1 ) 2/ p
2
x
+ O(x 1/ p−1/ p ),
p(2 p −1 )
p ≥ 3.
(n,m)∈Z2 \(0,0)
Unfortunately the l p ball and the underlying ellipse seem highly unsympathetic
to analysis, and we leave as an open question what one can provide in the way of
lower or upper bounds on σ0 in this case (the most natural example to consider
being p = 4).
8.5 Some analytic continuations
We shall write α(s) for the alternating zeta function (also known as the Dirichlet
eta function),
α(s) :=
∞
(−1)n+1
n=1
ns
,
and L ±d (s) for the L-series
L ±d (s) :=
∞
(±d|n)n −s ,
n=1
where (d|n) is the Kronecker (generalized Legendre) symbol.
258
Madelung sums in higher dimensions
When Q(x, y) := x 2 + py 2 with p = 3 or 7 we can write down an analytic
continuation of our lattice sum in terms α(s) and L − p (s):
(x, y) ∈ Z2 \ (0, 0)
(x, y) ∈ Z2 \ (0, 0)
(−1)x+y
= −2(1 + 21−s )α(s)L −3 (s),
(x 2 + 3y 2 )s
(−1)x+y
= −2α(s)L −7 (s),
(x 2 + 7y 2 )s
resembling the representation (see Glasser and Zucker [7])
(x, y) ∈ Z2 \ (0, 0)
(−1)x+y
= −4α(s)L −4 (s).
(x 2 + y 2 )s
These arise from our ability to write
(x, y) ∈ Z2 \ (0, 0)
for the quadratic form
1
= u ζ (s)L − p (s)
Q p (x, y)s
(8.5.1)
p+1
p+1
p−1
x2 +
xy +
y2
4
2
4
√
when p = 3, 7, 11, 19, 43, 67, or 163 (the primes for which Q( − p) is a unique
factorization domain), where u = 6 if p = 3 and 2 otherwise (the number of
√
units in Q( − p)). Unfortunately it is not clear how to insert a factor (−1)x+y
into (8.5.1), or how to replace the Q p (x, y) in those sums by x 2 + py 2 , other than
when p = 3 or 7.
Many sums of this type have been obtained by Glasser, Zucker, and Robertson
[7, 24, 25] for forms whose discriminant is disjoint (i.e., have one form per genus):
Q p (x, y) :=
(x, y) ∈ Z2 \ (0, 0)
(x, y) ∈ Z2 \ (0, 0)
(−1)x+y
1−t
=
−2
(1 − (2|μ)21−s )L ±μ L ∓4P/μ ,
(x 2 + P y 2 )s
μ|P
(−1)x
(x 2 + 2P y 2 )s
= −21−t
(1 − (2|μ)21−s )L ±μ L ∓8P/μ ,
μ|P
where L ±μ is taken such that μ ≡ ±1 (mod 4) and where the P’s are certain
square-free numbers (≡ 1 (mod 4) in the second case) with t prime factors, the
appropriate P’s less than 10000 being P = 5, 13, 21, 33, 37, 57, 85, 93, 105, 133,
165, 177, 253, 273, 345, 357, 385, 1365 and P = 1, 3, 5, 11, 15, 21, 29, 35, 39,
51, 65, 95, 105, 165, 231 respectively (in the latter case similar representations
can be shown to hold for x 2 + 8P y 2 ). Zucker and Robertson also obtained results
for the forms x 2 + P y 2 , x 2 + 4P y 2 , and x 2 + 16P y 2 when P = 3, 7, or 15
8.6 Some specific sums
259
(thus including the continuations of our sums above, although their approach is
different from ours).
8.6 Some specific sums
We have now obtained from the above theorems very explicit if quite contrasting
results regarding the range of convergence for shapes such as circles, diamonds
and squares. We continue with some related examples.
(1) It was shown in Borwein, Borwein, and Taylor [4, Section VI] that the study
of the Madelung constant for a two-dimensional hexagonal lattice with ions of
alternating unit√ charge placed at the lattice points of a lattice with basis vectors
(1, 0) and ( 12 , 23 ) leads naturally to sums of the form
(n,m)∈C N
(n 2
q(n, m)
,
+ nm + m 2 )s
where
q(n, m) :=
4
3
sin[ 23 (n + 1)π ] sin[ 23 (m + 1)π ] −
4
3
sin 23 nπ sin[ 23 (m − 1)π ].
Applying the above theorems, it is clear, on splitting q up appropriately, that
when Re s > 12 convergence occurs in such a sum when the lattice points are
summed over any vertically convex set, that for expanding rectangles convergence
holds for all Re s > 0, and that, when summing over expanding ellipses m 2 +
mn + n 2 ≤ N , convergence fails at some point between σ = 14 and σ = 23
73 .
Notice that (in the notation of Section 8.4.1)
1.
r ∗ (n) = 92
x 2 +x y+y 2 =n
x≡y≡0(mod 3)
This sum was also shown to possess a similar analytic continuation to those
mentioned in Section 8.5:
h 2 (s) = 3(1 − 31−s )(1 − 21−s )−1 α(s)L −3 (s).
(2) In [7], sums like
c2 (s) :=
( j,k)∈Cν
(−1) j
( j 2 + k 2 )s
and
c3 (s) :=
( j,k, p)∈Cν
(−1) j
( j 2 + k 2 + p 2 )s
260
Madelung sums in higher dimensions
are discussed; again they are covered by our previous analysis, with convergence
over vertically convex sets holding for all σ > 12 (respectively 1) and over circles
1
(respectively spheres) but failing at some point between 14 and 23
73 (respectively 2
and 34 ). After a change of variables j = j + k (respectively j = j + k + p),
convergence over squares (respectively cubes) can be seen (from Section 4.2) to
fail at 12 (respectively 1). However (from Section 4.3) convergence does hold for
all σ > 0 over certain other parallelepipeds.
(3) Let r N (n) denote the number of representations of n as a sum on N squares
(counting each permutation and sign change as a separate representation). Then
the Dirichlet sum
∞
r N (n)
(−1)n s
b N (s) :=
n
n=1
is a special case of the sums covered in Theorem 8.3 and Theorem 8.4. In
particular, it follows from [5, p. 290] that
b4 (s) := −8α(s)α(s − 1).
Notice that from our prior analysis the abscissa of convergence is
exactly equal to 1.
Correspondingly, b3 ( 12 ) is the Madelung constant for sodium chloride.
Theorem 8.9 recovers the fact that the limit is taken appropriately if we sum over
hypercubes. Theorem 8.2 shows that diamonds fail below 1, and Theorems 8.3
and 8.4 show that the exact abscissa for convergence over spheres lies between 12
and 34 . Theorem 8.4 recaptures the argument in [4] that shows that convergence
fails at 12 , and we would conjecture that convergence obtains for σ > 12 .
8.7 Direct analysis at s = 1
In the most basic case, Q(x, y) := x 2 + y 2 and s := 1, one can directly establish
that the limit A(C, Q, 1) = −π log 2 when C is the square |x| ≤ 1, |y| ≤ 1, the
diamond |x| + |y| ≤ 1, or the circle x 2 + y 2 ≤ 1 (i.e., summing over the standard
l p balls for p = 1, 2, or ∞) as we show below.
Observe that, when C is the above unit circle,
A(C, Q, 1) = lim
n→∞
∞
0<i 2 + j 2 ≤n 2
(−1)k r2 (k)
(−1)i+ j
,
=
2
2
k
i +j
k=1
where r2 (k) is the number of ways of expressing k as the sum of two squares of
integers in Z. Let
Sn :=
n
(−1)k r2 (k)
k=1
k
,
An :=
n
k=1
r2 (k).
8.7 Direct analysis at s = 1
261
Then
S2n = 2
n
r2 (2k)
k=1
2k
−
2n
r2 (k)
k=1
k
=−
2n
r2 (k)
,
k
k=n+1
since r2 (2k) = r2 (k); the latter follows because k = i 2 + j 2 ⇔ 2k = (i + j)2 +
(i − j)2 . It follows by partial summation that
S2n = −
2n
k=n
A2n
An
Ak
−
+
.
k(k + 1) 2n + 1
n
Further, it is familiar that
An
= π + n
n
with n = O(n −1/2 ).
Hence
S2n = −π
2n
k=n
k
1
−
+ o(1) = −π log 2 + o(1)
k+1
k+1
2n
as n → ∞.
k=n
Finally, since r2 (n) = O(n 1/2 ), we see that S2n −S2n−1 = o(1) and therefore that
lim Sn =
n→∞
∞
(−1)k r2 (k)
k=1
k
= −π log 2.
This shows that the method of expanding circles yields −π log 2 as the value of
the lattice sum
(−1)i+ j
.
i2 + j2
2
2
i + j >0
We show next that the method of expanding diamonds yields the same value.
This amounts to proving that
Tn :=
0<|i|+| j|≤n
(−1)i+ j
→ −π log 2
i2 + j2
as n → ∞.
Observe that, for 0 < t < 1,
f (t) :=
∞
k=1
r2 (k)t
k−1
2
2
∞
∞
∞
2
4 i2
1
1
4 i2
i
=
t −1 − =
t
−
t
2
t
t
t
t
i=0
and hence, by what has been proved above, that
1
f (−t) dt = π log 2.
0
i=0
i=0
262
Madelung sums in higher dimensions
But we also have
∞
− f (−t) = 4
⎛
(−1)k ⎝
k=0
∞
=4
k
⎞
t
(k− j)2 + j 2 −1
−t
k 2 −1
⎠
j=0
(−1)k
k=1
k−1
t (k− j)
2 + j 2 −1
,
j=0
so that
1
−
∞
k−1
(−1)k
f (−t) dt = 4
0
k=1
j=0
1
= lim Tn ,
n→∞
(k − j)2 + j 2
provided we can justify the term-by-term integration. This can be done as follows:
Note that, for 0 < t < 1,
t
−
∞
f (−u) du = 4
(−1)k δk (t),
0
k=1
where
δk (t) :=
k−1 (k− j)2 + j 2 −1
t
,
(k − j)2 + j 2
j=0
and that, for 0 ≤ j ≤ k,
k2
≤ (k − j)2 + j 2 ≤ k 2 .
2
Further, for 0 < t < 1, k ≥ 2,
δk−1 (t) − δk (t) =
k−1
j=1
t (k− j) + j −1
t (k− j) + j −1
−
2
2
(k − j) + ( j − 1)
(k − j)2 + j 2
2
2
2
2
2
2
k−1 (k− j)2 −1 ( j−1)2
t
(t
− t j ) t k −1
+
− 2 ,
(k − j)2 + ( j − 1)2
k
j=1
so that
|δk−1 (t) − δk (t)| ≤
k−1 j=1
+
k−1
j=1
1
1
−
(k − j)2 + ( j − 1)2
(k − j)2 + j 2
t ( j−1) − t j
1
+ 2
2
2
(k − j) + ( j − 1)
k
2
2
8.7 Direct analysis at s = 1
≤
k−1
j=1
+
263
j 2 − ( j − 1)2
1
2
2
(k − j) + ( j − 1) (k − j)2 + j 2
k−1
1
2 ( j−1)2
2
t
−tj + 2
2
k
k
j=1
≤
k−1 2
4
1
7
2
2
j
+ 2 + 2 = 2.
−
(
j
−
1)
2
2
k (k − 1)
k
k
k
j=1
Also, for 0 < t < 1,
0 ≤ δk (t) ≤
k−1
j=0
1
2k
2
≤ 2 = →0
k
(k − j)2 + j 2
k
as k → ∞.
It follows, by the Weierstrass M-test, that
4
∞
(−1)k δk (1) = − lim 4
t→1−
k=1
= − lim
∞
[δ2k−1 (t) − δ2k (t)]
k=1
t
t→1− 0
1
f (−u) du = −
f (−u) du,
0
and this completes the proof of the expanding diamonds case.
Finally we shall show that the method of expanding squares also yields the
value −π log 2 for the lattice sum. In fact we shall deal with the slightly more
general method of expanding rectangles. Let
(−1)i+ j
2
2
,
R
(t)
:=
t i + j −1 .
Rm,n :=
m,n
2
2
i +j
|i|+| j|>0
|i|+| j|>0
|i|≤n,| j|≤m
|i|≤n,| j|≤m
We shall prove that Rm,n → −π log 2 when μ := min(m, n) → ∞. Observe that
1
Rm,n = −
Rm,n (−t) dt
0
and that, for 0 < t < 1,
⎞
⎛
1
2
2
2
2
2
2
f (t) − Rm,n (t) = ⎝
ti + j +
ti + j +
ti + j ⎠
t
|i|>n,| j|>m
|i|≤n,| j|>m
|i|>n,| j|≤m
⎞
⎛
∞
∞
n
∞
∞
m
4
2
2
2
2
2
2
= ⎝
ti
tj +
ti
tj +
ti
tj ⎠.
t
i=n+1 j=m+1
Observe also that, for 0 < t < 1,
∞
2
2
i (−t) < t (n+1)
i=n+1
i=0
and
j=m+1
i=n+1
n
2
(−t)i < 1.
i=0
j=0
264
Madelung sums in higher dimensions
Hence, for 0 < t < 1,
| f (−t)−Rm,n (−t)| < 4t (n+1)
and so
|Rm,n
2 +(m+1)2 −1
+4t (m+1)
2 −1
+4t (n+1)
2 −1
< 12t (μ+1)
2 −1
,
1
1
12
+ π log 2| = f (−t) dt −
Rm,n (−t) dt ≤
→0
0
(μ + 1)2
0
as μ → ∞.
This completes the proof.
8.8 Proofs
Proof of Theorem 8.1
We first note that for a positive definite quadratic form
Q(x1 , . . . , xk ) =
k k
αi j xi x j ∈ R[x1 , . . . , xk ],
i=1 j=1
with αi j = α ji , we have
Q(x1 , . . . , xk ) ≥ λ(x12 + · · · + xk2 )
for some λ = λ Q > 0. Suppose in what follows that σ > 12 (k − 1), ν > 0, and
the ji are integers. Let
Q( j1 , . . . jk , s) := Q( j1 , . . . , jk )−s
Q(0, . . . , 0, s) := 0,
when j12 + · · · + jk2 = 0,
and let
μ :=
sup
ν>0
( j1 ,..., jk )∈Zk
|wν ( j1 , . . . , jk )| < ∞.
Then we have
Aν (s) =
=
∞
···
∞
q( j1 , . . . , jk )χν ( j1 , . . . , jk )Q( j1 , . . . , jk , s)
j1 =−∞ jk =−∞
∞
∞
···
wν ( j1 , . . . , jk )[Q( j1 , . . . , jk−1 , jk , s)
j1 =−∞ jk =−∞
−Q( j1 , . . . , jk−1 , jk + 1, s)]
∞
∞
=:
···
a j1 ,..., jk (ν, s).
j1 =−∞ jk =−∞
8.8 Proofs
265
Suppose next that σ > 12 (k − 1) + > 12 (k − 1). Observe that, for 0 < u < v,
v
v−u
1
−1−s
− 1 = s
t
dt ≤ 1+σ |s|
us
vs u
u
and hence that, when ( j12 + · · · + jk2 )( j12 + · · · + ( jk + 1)2 ) = 0,
|a j1 ,..., jk (ν, s)| ≤
=
≤
|Q( j1 , . . . , jk ) − Q( j1 , . . . , jk + 1)|
|s|μ
λ1+σ ( j12 + · · · + jk2 )1+σ
k−1
αik ji + αkk (2 jk + 1)|
|2 i=1
|s|μ
λ1+σ ( j12 + · · · + jk2 )1/2
( j12
λ−1−σ |s|M
,
+ · · · + jk2 )σ +1/2
where
M :=
( j12 + · · · + jk2 )σ +1/2
|2
sup
k−1
i=1
αik ji + αkk (2 jk + 1)|
( j12
j12 +···+ jk2 >0
+ · · · + jk2 )1/2
μ < ∞.
Also, when ( j12 + · · · + jk2 )( j12 + · · · + ( jk + 1)2 ) = 0,
|a j1 ,..., jk (ν, s)| ≤
μ
.
|αk,k |σ
Since (as in Borwein, Borwein, and Taylor [4])
( j2
j12 +···+ jk2 >0 1
1
< ∞,
+ · · · + jk2 )(k/2)+
it follows, by the Weierstrass M-test, that, when ν → ∞, Aν (s) → A(s), say,
uniformly in the region {s : σ > 12 (k − 1) + , |s| ≤ K } with K any fixed
positive number and, since Aν (s) is analytic in this region, that A(s) is analytic
therein. Consequently
A(s) = lim An (s)
n→∞
exists and is analytic in the region Re s > 12 (k − 1).
Proof of Theorem 8.2 We suppose that C is the diamond a1 |x1 | + · · · + ak |xk | ≤
c where d = gcd(a1 , . . . , ak ) and that all the ai /d are odd positive integers.
Observing that when
a1 |x1 | + · · · + ak |xk | = cn
and n is a multiple of d we have
(−1)x1 +···+xk = (−1)cn/d
266
Madelung sums in higher dimensions
and
⎛
|Q(x1 , · · · , xk )| ≤ ⎝
k k
⎞
|αi, j |⎠ c2 n 2 ,
i=1 j=1
it is not hard to see that, when s = 12 (k − 1) and N is a multiple of B := a1 · · · ak ,
A N (s) − A N −1/c (s) =
Q(x1 , . . . , xk )−(k−1)/2
a1 |x1 |+···+ak |xk |=cN
>
1
c1 N k−1
1
c1 N k−1
> c2 > 0
≥
1
|x1 |+···+|xk |=(cN /B)
(cN /B) + k − 1
k−1
as N → ∞ (where in fact we have only bothered to count the points with xi =
(B/ai )xi ≥ 0); and the limit cannot exist.
Proof of Corollary 8.1
Under the hypothesis given, it is simple to estimate that
m
wν ( j1 , . . . , jk−1 , m) :=
q( j1 , . . . , jk−1 , l)χν ( j1 , . . . , jk−1 , l)
l=−∞
is uniformly bounded.
Proof of Theorem 8.3 Suppose that q(x1 , . . . , xk ) is periodic with period M;
then (with S0 (x) as defined in (8.4.1)), dividing the sum into residue classes
modulo M we have
S0 (x) =
M
r1 =1
···
M
q(r1 , . . . , rk )F0 (r1 , . . . , rk , x),
(8.8.1)
rk =1
where
F0 (r1 , . . . , rk , x) :=
1.
Q(x1 ,...,xk )≤x
xi ≡ri mod M
Writing xi = M yi + ri it is easy to see that F(r1 , . . . , rk , x) counts the number
of lattice points (y1 , . . . , yk ) ∈ Zk in the expanded ellipse x 1/2 E, where E is the
ellipse M 2 Q(y1 , . . . , yk ) ≤ 1 of area
A=
π k/2
√
M k |D| ( 12 k + 1)
with its centre shifted to (−r1 /M, . . . , −rk /M). Approximating the number of
points in the ellipse by its area we can write
F0 (r1 , . . . , rk , x) = Ax k/2 + O(ψ),
8.8 Proofs
267
where from the results of Huxley [9, Theorem 5] (for k = 2), Krätzel and Nowak
[11] (for k = 3), Walfisz [20], and Landau [12, 13] (for k ≥ 8 and k ≥ 4 respectively; see Landau [14, Satz II] for the most immediately applicable form), we can
take
⎧ 23/73
x
(log x)315/146 if k = 2,
⎪
⎪
⎨ 25/34
(log x)10/17
if k = 3,
x
ψ=
2
⎪
x
if k = 4,
x
log
⎪
⎩ k/2−1
if k ≥ 5.
x
Hence
S0 (x) = B1 x k/2 + O (B2 ψ) ,
where
B1 :=
M
···
r1 =1
M
q(r1 , . . . , rk ),
B2 :=
rk =1
M
···
r1 =1
M
|q(r1 , . . . , rk )|,
rk =1
and the result is plain from (8.4.2).
Proof of Theorem 8.4 Notice that if Q(x1 , . . . , xk ) = a1 x12 + · · · + ak xk2 with all
the ai odd positive integers and q(x1 , . . . , xk ) = (−1)x1 +···+xk then
1.
r (n, Q, q) = (−1)n
Q(x1 ,...,xk )=n
Now, by elementary methods we have
|r (n, Q, q)| =
1 = A[1 + o(1)]x k/2
Q(x1 ,...,xk )≤x
n≤x
as x → ∞, where A is the area of the ellipse Q(x1 , . . . , xk ) ≤ 1. In particular
there must certainly be infinitely many integers n with |r (n, Q, q)| > 12 An k/2−1 .
Hence, if Re s ≤ k/2 − 1 then
|A N (s) − A N −1 (s)| =
|r (N , Q, q)|
→ 0
Nσ
as N → ∞, and the limit A(s) cannot exist.
Proof of Theorem 8.5 We closely follow the proof of the corresponding result
for the error in the classical circle problem (as given in Landau [15]) and for ρ ≥ 1
inductively define
x
Sρ (u) du,
Sρ+1 (x) :=
0
where S0 (x) is as defined in (8.4.1) (so that equivalently
x
1 1
(x − n)ρ r (n, Q, q) =
S0 (u)(x − u)ρ−1 du
Sρ (x) =
ρ!
(ρ − 1)! 0
0≤n≤x
268
Madelung sums in higher dimensions
for all ρ ≥ 1). We invoke the following lemma of Riesz [18] (as in Wilton [23];
cf. Landau [15, Satz 533]).
Lemma 8.1
and x > 0,
If f 0 (x) is L-integrable and bounded over (0, x), if, when γ > 0
1
f γ (x) :=
(γ )
x
f 0 (u)(x − u)γ −1 du,
0
and if V (x) and W (x) are increasing functions of x with
| f 0 (x)| < V (x)
and
| fl (x)| < W (x)
then
| f β (x)| < b(β, l)V (x)(1−(β/l)) W (x)(β/l)
for all 0 ≤ β ≤ l, where the b(β, l) depend only on β and l.
We shall show that (as long as Aν (s) is not identically zero) there is a positive
integer ν and non-zero constants Bν+1 and Cν such that
|Sν+1 (x)| < Bν+1 [1 + o(1)]x (k+2(ν+1)−1)/4
for all x and
|Sν (x)| > |Cν |[1 + o(1)]x (k+2ν−1)/4
for infinitely many x.
Hence, by the above lemma we must have
ν+1
|Cν |
|S0 (x)| ≥ [1 + o(1)]
x (k−1)/4
ν/(ν+1)
b(ν, ν + 1)Bν+1
for infinitely many x.
Thus it remains to justify the claimed upper and lower bounds for the Sρ (x).
We define the constants
√
∞
π
r ∗ (n)
( D M)(k+2ρ−1)/2 cos(k + 2ρ + 1) ,
Cρ :=
4
Mπ ρ+1
n (k+2ρ+1)/4
n=1
√
∞
( D M)(k+2ρ−1)/2 |r ∗ (n)|
Bρ :=
.
Mπ ρ+1
n (k+2ρ+1)/4
n=1
Lemma 8.2 We suppose that q(x1 , . . . , xk ) is periodic in all the xi with period
M and
M
M
···
q(r1 , . . . , rk ) = 0.
r1 =1
rk =1
8.8 Proofs
269
For all ρ > 12 (k − 1) we have
|Sρ (x)| ≤ Bρ 1 + O(x −1/2 ) x (k+2ρ−1)/4
for all x and, if Cρ = 0,
log x −1/4
|Sρ (x)| = Cρ 1 + O
x (k+2ρ−1)/4
log log x
for infinitely many integers x.
Proof For ρ ≥ 0 we define inductively
r , x) :=
Fρ+1 (
x
0
Fρ (
r , u) du
and observe (by repeated integration of (8.8.1)) that
Sρ (x) =
M
···
r1 =1
M
q(r1 , . . . , rk )Fρ (r1 , . . . , rk , x).
rk =1
It has been shown by a number of authors (see for example Landau [12]) that the
Fρ (
r , x) can be expressed in terms of Bessel functions:
√
( D M)(k+2ρ−2)/2
r , x) = Vρ (x) +
E(
r , x),
Fρ (
Mπ ρ
where
π k/2
Vρ (x) :=
x k/2+ρ
M k D ( 12 k + ρ + 1)
and
E(
r , x) := x k/4+ρ/2
∞
2π
r ∗ (n; r)
J
k/2+ρ
M
n k/4+ρ/2
n=1
with
∗
r (n; r) :=
x∈Zk
Q ∗ (
x )=n
(
nx
D
2π
r · x .
cos
M
Notice that, from the straightforward bounds
1 = O(z k/2 ),
Jν (z) = O(z −1/2 ),
x∈Zk
Q ∗ (
x )≤z
such a sequence is absolutely convergent for ρ > 12 (k − 1).
Hence, if
M
r1 =1
···
M
rk =1
q(r1 , . . . , rk ) = 0
,
270
Madelung sums in higher dimensions
then we obtain
√
( ∞
2π nx
( D M)(k+2ρ−2)/2 k/4+ρ/2 r ∗ (n)
Sρ (x) =
.
x
Jk/2+ρ
ρ
k/4+ρ/2
Mπ
M D
n
n=1
Approximating the Bessel functions by cosines (see for example Watson [22,
p. 199]), so that
(
2
π
+ O(z −3/2 ),
Jν (z) =
cos z − (2ν + 1)
πz
4
we obtain
√
( D M)(k+2ρ−1)/2 (k+2ρ−1)/4
Sρ =
x
(M1 + M2 )
Mπ ρ+1
with
( ∞
π
r ∗ (n)
2π nx
− (k + 2ρ + 1)
cos
M1 :=
(k+2ρ+1)/4
M D
4
n
n=1
and
M2 := O x
−1/2
∞
n=1
(the latter bound follows since
r ∗ (n)
n (k+2ρ+3)/4
= O x −1/2
|r ∗ (n)| = O(z k/2 )
(8.8.2)
n≤z
and by assumption 2ρ + 1 > k). The trivial bounding
|M1 | ≤
∞
n=1
|r ∗ (n)|
n (k+2ρ+1)/4
gives us the required upper bound.
By the box principle (in k dimensions), given an N and also k real numbers
ν1 , . . . , νk , there is certainly an integer 1 ≤ m ≤ N k + 1 such that the distances
from the mνi to their nearest integers simultaneously satisfy ||mνi || < 1/N .
√
N
In particular
√ (taking k = N ) there is an integer m = z ≤ N + 1, with
√
m( n/M D) close enough to an integer for n = 1 to N that
(
π
π
1
2π nz
− (k + 2ρ + 1) = cos(k + 2ρ + 1) + O
cos
M |D|
4
4
N
log log 16z
π
= cos (k + 2ρ + 1) + O
4
log 16z
for all n = 1, . . . , N ; the latter equality follows from the observation that, since
log 16z > e,
log 16(N N + 1)2
log 16z
≤
= O(N ).
log log 16z
log log 16(N N + 1)2
8.8 Proofs
271
Using the estimate (8.8.2) we can readily bound the remaining terms in the
sum by
∞
n=N
r ∗ (n)
n (k+2ρ+1)/4
= O(N −(2ρ+1−k)/4 )
= O(N
−1/4
)=O
log log 16z
log 16z
1/4 .
Hence, for such a z, as long as Cρ = 0 we have
∞
r ∗ (n)
π
log log 16z 1/4
,
M1 = cos (k + 2ρ + 1)
1+O
(k+2ρ+1)/4
4
log 16z
n
n=1
and the remaining bound is plain. Varying N (and hence the closeness of the
approximation) we can clearly generate infinitely many integers z in this way.
Final step of the proof of Theorem 8.5 Hence it only remains to justify that (as
long as r (n, Q, q) is not identically zero) Cρ is non-zero for some ρ. First, observe
that r ∗ (n) cannot be identically zero. If r ∗ (n) were identically zero then we would
have Sρ (x) = 0 for all x ≥ 0 and any ρ > 12 (k − 1); in particular it would follow
from the relation
r (N , Q, q) = (N + 1)! Sρ (N + 1) −
N
−1
(N + 1 − n)ρ r (n, Q, q)
n=0
and an easy induction on N that r (n, Q, q) would have to be identically zero.
We suppose that w is the smallest positive integer such that r ∗ (w) = 0. From
the lower bound Q ∗ (x1 , . . . , xk ) > λ∗ (x12 + · · · + xk2 ) we certainly have the trivial
lower bound
⎛
⎞
M
M
2 k
∗
k/2
⎝
⎠
|r (n)| ≤ Bn ,
B :=
···
|q(r1 , . . . , rk )|
.
√
λ∗
r =1
r =1
1
Hence if
R ≥ N := max
we have
k
7
2
1
π
B
k
k/2+2
log
1
+
+ 2 , log
(w
+
1)
2
12 |r ∗ (w)|
w
∞
∞
k/2+2 r ∗ (n) r ∗ (w) B(w
+
1)
1
<
≤
wR ,
R (w + 1) R
n2
n=w+1 n n=1
∗
R
and ∞
one of ρ = [(4N − k)/2] + 1 and
n=1 r (n)/n = 0. In particular at least
1
∗
(k+2ρ+1)/4 and cos[(k +
ρ = [ 2 (4N − 1 − k)] + 1 will have both ∞
n=1 r (n)/n
2ρ + 1)π/4] non-zero, and hence Cρ = 0 .
272
Madelung sums in higher dimensions
Proof of the representation (8.4.4) By partial summation and integration by parts
we obtain
ν
S0 (ν) − q(0)
d (u −s ) du
Aν (s) =
−
[S0 (u) − q(0)]
νs
du
1
=
ρ
(s + i) Si (ν)
i=0
=
ρ−1
i=0
(s)
(s + i)
(s)
ν s+i
(s + ρ + 1) ν
− Si (1) +
Sρ (u)u −s−ρ−1 du
(s)
1
Si (ν)
(s + ρ)
Sρ (1)
− Si (1) −
ν s+i
(s)
(s + ρ + 1)
+
(s)
∞
Sρ (u)u −s−ρ−1 du + O(ν −(ρ−(k−1)/2+2σ )/2 ),
1
for bounded |s|, since by Lemma 8.2
Sρ (u) = O(u (k+2ρ−1)/4 )
for ρ > 12 (k − 1).
Proof of Theorem 8.6
Writing
r (n, r) = r (n; r1 , . . . , rk ) :=
1,
Q(x1 ,...,xk )=n
xi ≡ri (mod M)
for Re s > 0 we define the theta function
θ̂(s) :=
∞
r (n, Q, q)e−ns
n=0
=
M
r1 =1
···
M
q(r1 , . . . , rk )θ̂ (s; r )
rk =1
where
θ̂ (s; r ) :=
∞
r (n; r )e−ns .
n=0
Now, by Novák [16, Lemma 1] for integers h and l (with (h, l) = 1 and l > 0)
the modular functions θ̂ (s; r) can be expanded in a neighbourhood of the cusp
8.8 Proofs
273
2πi h/l, so that, for Re s > 0,
θ̂ (s; r) =
π k/2 (s − 2πi h/l)−k/2
√
D M k/2l k
×
Sh,l (m,
r) exp −
m
∈ Zk
where
Sh,l (m,
r ) :=
l
···
a1 =1
l
ak =1
π 2 Q ∗ (m)
,
D M 2l 2 (s − 2πi h/l)
2πi
2πi h
Q(
a M + r ) +
m
· (
a M + r ) .
exp −
l
lM
Thus
2πi h
π k/2
r) + E h,l (
Sh,l (0,
r, σ) ,
θ̂ σ +
, r = √
l
D M k/2l k σ k/2
where, for fixed h and l,
π 2 λ∗
k
2
2
E h,l (
(m + · · · + m k )
r, σ) ≤
l exp −
D M 2l 2 σ 1
k
m∈
Z
m
=0
=O
∞
n
k/2 −cn/σ
e
= O(e−c/σ ).
n=1
Now, if we are in the non-singular case we can pick h and l such that
A := √
M
π k/2
D M k/2l k
···
r1 =1
M
r) = 0
q(r1 , . . . , rk )Sh,l (0,
rk =1
(notice that, since the q(r1 , . . . , rk ) sum to zero, h = 0) and hence
2πi h
= A + O(e−c/σ )
σ k/2 θ σ +
l
uniformly in σ . With S0 (x) as in (8.4.1), writing
∞
θ̂ (s) = s
e−xs S0 (x) d x,
0
for all x would imply that
it is clear that |S0 (x)| <
∞
2πi h
k/2
k/2
< c |σ + (2πi h/l)| σ
σ θ̂ σ +
e−σ x x k/2−1 d x
l
0
cx k/2−1
= c |σ + (2πi h/l)| ( 12 k).
Hence, on letting σ → 0, we see that, for any constant
c1 < A(2π h/l)−1 (( 12 k))−1 ,
we must have |S0 (x)| > c1 x k/2−1 for infinitely many integers x.
274
Madelung sums in higher dimensions
Proof of Corollary 8.2
We take h = 1 and l to be a high power of M:
l := M γ ,
γ > 2α,
where α is the highest power of a prime factor of M dividing 2k D (recall that
D, defined in (8.4.3), is the determinant of the matrix of coefficients αi j of
Q(x1 , . . . , xk )).
Writing
R := {(x1 , . . . , xk ) ∈ Zk : 1 ≤ xi ≤ l}
and
2πi
Q(
x ) q(
x ),
F(
x ) := exp −
l
non-singularity will follow once we show the non-vanishing of
S := M −k
lM
x1 =1
Now, we have
|S|2 =
lM
···
F(
x) =
F(
x ).
x∈R
xk =1
F(y )F(
x)
x∈R y∈R
=
F(
u + x)F(
x)
x∈R u∈R
=
F(
u)
u∈R
⎧
k ⎨
l
i=1
⎩
xi =1
⎫
k
⎬
2πi xi 2
exp −
αi j u j
⎭
l
j=1
on noting that
q(
u + x) = q(
u )q(
x)
and making the expansion
Q(
u + x) − Q(
x ) = Q(
u) +
k
i=1
⎛
xi ⎝2
k
⎞
αi j u j ⎠ .
j=1
Observing that
l
xi =1
and setting
L :=
⎧
⎨
⎩
2πi xi α
=
exp −
l
(u 1 , . . . , u k ) ∈ R : 2
k
j=1
l
0
if α ≡ 0 (mod l),
otherwise,
αi j u j ≡ 0 (mod l), 1 ≤ i ≤ k
⎫
⎬
⎭
,
8.8 Proofs
we obtain
|S|2 = l k
275
F(
u ).
u∈L
It is not hard to check that any u satisfying the linear system in L must necessarily
satisfy
2k Du i ≡ 0 (mod l),
1 ≤ i ≤ k,
u i ≡ 0 (mod M γ −α ),
1 ≤ i ≤ k.
and therefore certainly
Since we have chosen γ > 2α we thus have
q(
u ) = 1,
Q(
u ) ≡ 0 (mod l),
for any u ∈ L, giving
|S|2 =
1 = 0
u∈L
(plainly (l, . . . , l) is in L) and we are in the non-singular situation for any
Q(x1 , . . . , xk ).
Proof of Proposition 8.1 (see Section 8.4) Clearly it is enough to show the result
for triangles T , and in fact (by taking sums and differences) enough to consider
triangles with one vertex at the origin. Using the symmetries for x → −x, y →
−y we shall further assume that the triangle T lies entirely in the quadrant x, y ≥
0 and (replacing N by cN or N /gcd(a, b) as necessary) that T takes the form
T = {(x, y) : r2 x ≥ s2 y, r1 x ≤ s1 y, ax − by ≤ 1},
where a, b, si , ri ∈ Z with ri , si ≥ 0 and gcd(a, b) = gcd(ri , si ) = 1. We denote
the sides of T by li
l1 : r1 x = s1 y,
l2 : r2 x = s2 y,
l3 : ax − by = 1,
and the points of intersection of l3 with l1 and l2 by P1 and P2 respectively:
si
ri
.
Pi =
,
asi − bri asi − bri
Choosing integers x0 , y0 satisfying
ax0 − by0 = 1
and writing
Ai = ri x0 − si y0 ,
Bi = asi − bri ,
αi =
Ai
Bi
276
Madelung sums in higher dimensions
(notice that gcd(Ai , Bi ) = 1 and Bi > 0), we can parameterize the integer points
on n(l3 ∩ T ), the intersection of the line ax − by = n with N T by
x = nx0 + bt,
y = ny0 + at,
nα1 ≤ t ≤ nα2
for n = 1 to N , with (−1)x+y = (−1)n(x0 +y0 )+t (a+b) . We distinguish the two
cases (i) 2|ab, (ii) 2 |ab.
Case (i): a and b are not both odd. Since a and b are of opposite parity we
can pick x0 , y0 to both be odd (indeed, either (x0 , y0 ) or (x0 + b, y0 + a) will
be of this form). Hence on the line segment n(l3 ∩ T ) our parameterization gives
(−1)x+y = (−1)t and, writing
f n (t) := Q(nx0 + bt, ny0 + at)−s ,
we have
A N (T, Q, s) =
An (l3 ∩ T, Q, s),
1≤n≤N
where
An (l3 ∩ T, Q, s) =
(−1)t f n (t).
nα1 ≤t≤nα2
We first observe some elementary bounds on f n :
| f n (t)| = |Q(x, y)−s | = O(n −2σ ),
| f n (t)| = | − s (2αb + βa)x + (2γ a + βb)y Q(x, y)−s−1 | = O(n −2σ −1 ),
(s + 1)
Q(a, b) 2
| f n (t)| = (2αb + βa)x + (2γ a + βb)y
−
2s
Q(x, y)s+2
Q(x, y)s+2 = O(n −2σ −2 ).
(8.8.3)
Pairing odd and even t we have
An (l3 ∩ T, Q, s) = M1 + M2 ,
where
M1 :=
f n (2t) − f n (2t + 1)
1
1
2 nα1 ≤t≤ 2 nα2
and
M2 := u 2 (n) f n (2tn,2 + 1) − u 1 (n) f n (2tn,1 + 1)
with u 1 (n) := 1 if there is an integer tn,1 in n2BA11 − 12 , n2BA11 and 0 otherwise, and
u 2 (n) := 1 if there is an integer tn,2 in n2BA22 − 12 , n2BA22 and 0 otherwise. Using
8.8 Proofs
277
the bound for f n (t) we have
f n (2t + 1) − f n (2t) =
2t+1
f n (u)du
2t
=
2t+1
2t
[ f n (2t) + O(n −2σ −2 )] du.
This and the observation that, for a differentiable function g(x),
g(n) = [x2 ]g([x2 ]) − [x1 ]g([x1 ]) −
x1 <n≤x2
=
[x2 ] %
[x1 ]
[x2 ]
[x1 ]
[u]g (u) du
&
g(u) − {u}g (u) du,
where [x] and {x} respectively denote the integer and fractional parts of x, enables
us to evaluate M1 :
M1 = − 12
1
1
2 nα1 ≤t≤ 2 nα2
d
f n (2t) + O(n −2σ −2 )
dt
d
−2σ −2
f
(2t)
+
O(n
)
dt + O(n −2σ −1 )
n
1
dt
2 nα1
= − 12 f n 2 × 12 nα2 − f n 2 × 12 nα1 + O(n −2σ −1 )
=
− 12
1
2 nα2
= 12 Q(n P1 )−s − 12 Q(n P2 )−s + O(n −2σ −1 ).
For M2 (approximating 2tn,i + 1 by nαi ) we have
u i (n) f (2ti,n + 1) = u i (n)Q(n Pi )−s + O(n −2σ −1 ),
giving
An (l3 ∩T, Q, s) = 12 Q(P1 )−s
1 − 2u 1 (n) 1
1 − 2u 2 (n)
− 2 Q(P2 )−s
+ O(n −2σ −1 ).
n 2s
n 2s
Notice that
An (l3 ∩ T, Q, s) = O(n −2σ ).
(8.8.4)
Hence
1 − 2u 1 (n)
1 − 2u 2 (n)
− 12 Q(P2 )−s
2s
n
n 2s
n≤N
n≤N
1
,
+ C1 (s) + O
N 2σ
A N (T, Q, s) = 12 Q(P1 )−s
278
Madelung sums in higher dimensions
where C1 (s) is analytic for all Re s > 0. Noting that the functions u i (n) are
defined modulo (2Bi ) and that, for a function u(n) ≤ 1 defined modulo q by
u(n)n −2s = (kq)−2s
u(n) + O(qk −2σ −1 )
kq≤n<(k+1)q
=
kq≤n<(k+1)q
1
q
q
u(m)
m=1
n −2s + O(qk −2σ −1 ),
kq≤n<(k+1)q
we can clearly write
⎛
⎞
2Bi
1 − 2u i (n)
1
−2s
⎝
⎠
+ C2,i (s) + O(N −2σ ).
= 1−
u i (m)
n
Bi
n 2s
1≤n≤N
m=1
1≤n≤N
with C2,i (s) analytic in Re s > 0. Now
2B1
u 1 (n) = #{1 ≤ n ≤ 2B1 : n A1 ≡ 1, 2, . . . , or B1 (mod 2B1 )}
m=1
=
B1
B1 − 1
if 2 |A1 ,
if 2|A1 ,
and
2B2
u 2 (n) = #{1 ≤ n ≤ 2B2 : n A2 ≡ 0, 1, . . . , or (B2 − 1) (mod 2B2 )}
m=1
=
B2
B2 − 1
if 2 |A2 ,
if 2|A2 .
Noting that (since x0 and y0 are both odd) 2|Ai exactly when 2 |ri si , we obtain
1 λ(l1 )
1 λ(l2 )
A N (T, Q, s) =
Q(P1 )−s
n −2s +
Q(P2 )−s
n −2s
2 B1
2 B2
+ C3 (s) + O(N
n≤N
−2σ
n≤N
),
where
λ(li ) =
1
0
if 2 |ri si ,
if 2|ri si ,
and C3 (s) is analytic for Re s > 0. Since we have already shown that
A N (l3 ∩ T, Q, s) = O(N −2σ )
it will be enough to show that, for i = 1, 2,
λ(li )
A N (li ∩ T, Q, s) =
Q(Pi )−s
n −2s + C4,i (s) + O(N −2σ ),
Bi
n≤N
8.8 Proofs
279
for some suitably analytic C4,i (s).
Since N (li ∩ T ) ∩ Z2 = {n Bi Pi : 1 ≤ n ≤ N /Bi } we have
A N (li ∩ T, Q, s) = Q(Bi Pi )−s
(−1)ri +si n −2s .
n≤N /Bi
Now, if λ(li ) = 0 we have ri + si odd and
A N (li ∩ T, Q, s) = Q(Bi Pi )−s
(−1)n n −2s
n≤N /Bi
= Q(Bi Pi )−s
=
(2t)−2s − (2t − 1)−s + O(N −2σ )
t≤N /2Bi
O(t
−2σ −1
) + O(N −2σ )
t≤N /Bi
= C4,i (s) + O(N −2σ ),
(8.8.5)
while if λ(li ) = 1 then ri + si is even and
Q(Pi )−s −2s
Q(Pi )−s
n
=
Bi
Bi
n≤N
=
Bi
(n Bi + l)−2s + O(N −2σ )
n≤N /Bi l=1
Bi
Q(Pi )−s (n Bi )−2s+
O(n −2σ −1 )+O(N −2σ )
Bi
n≤N /Bil=1
= Q(Bi Pi )−s
n≤N /Bi
n −2s + C5,i (s) + O(N −2σ ),
n≤N /Bi
where the C4,i (s) and C5,i (s) are analytic for Re s > 0, as required.
Case (ii): Both a and b are odd. When a and b are both odd, any x0 and y0
satisfying ax0 − by0 = 1 are necessarily of opposite parity, so that on the line
ax − by = n our parameterization gives (−1)x+y = (−1)n . We here choose our
x0 , y0 to satisfy Ai = ri x0 − si y0 > 0 for i = 1, 2 (we can do this by replacing
x0 , y0 by x0 + bj, y0 + a j for a suitably small j), and set
βi =
Bi
.
Ai
Hence, altering the order of the n and t summations, we have
A N (T, Q, s) =
(−1)n
f n (t)
n≤N
=
α1 ≤t≤N α1
nα1 ≤t≤nα2
E1 +
N α1 ≤t≤N α2
E2,
280
Madelung sums in higher dimensions
where
E 1 :=
(−1)n f n (t)
tβ2 ≤n≤tβ1
E 2 :=
(−1)n f n (t).
tβ2 ≤n≤N
Just as in case (i) (with the roles of n and t reversed), summing along the line
joining tβ1 P1 and tβ2 P2 we have
1 − 2v2 (t) 1
1 − 2v1 (t)
1
Q (β2 P2 )−s
− Q (β1 P1 )−s
+ O(t −2σ −1 )
2
2
t 2s
t 2s
−s
B2
1 − 2v2 (t) 1
1
1 − 2w(N )
E2 = Q
P2
− f N (t)
+ O(t −2σ −1 )
2
A2
2
t 2s
t 2s
t B1
t B1
and 0 otherwise, v2 (n) =
− 12 , 2A
where v1 (n) = 1 if there is an integer in 2A
1
1
t B2
t B2
and 0 otherwise, and w(N ) is 1 if N is
1 if there is an integer in 2A
− 12 , 2A
2
2
even and 0 if N is odd. Hence
1 − 2v2 (t)
1
1
A N (T, Q, s) =
(−1) N f N (t) + Q (β2 P2 )−s
2
2
t 2s
E1 =
N α1 ≤t≤N α2
α1 ≤t≤N α1
2v1 (t) − 1
1
+ Q (β1 P1 )−s
+ C6 (s) + O(N −2σ ),
2
t 2s
α1 ≤t≤N α1
with C6 (s) analytic in Re s > 0. It is not hard to see that the first sum is simply
1
2 A N (l3 ∩ T, Q, s). Hence it remains to verify that the other sums differ from
1
1
2 A N (l2 ∩ T, Q, s) and 2 A N (l1 ∩ T, Q, s) by a function analytic for Re s > 0.
One proceeds just as in case (i) (with the roles of Ai and Bi reversed and with
2|Bi if and only if 2 |ri si ), showing that, for i = 1, 2,
1 − 2vi (t)
(−1)i Q (βi Pi )−s
t 2s
αi ≤t≤N αi
λ(li )
Q (βi Pi )−s
t −2s + C7,i (s) + O(N −2σ )
Ai
αi ≤t≤N αi
−s
= λ(li )Q (Bi Pi )
t −2s + C8,i (s) + O(N −2σ )
=
1/Bi ≤t≤N /Bi
= A N (li ∩ T, Q, s) + C9,i (s) + O(N −2σ ),
with C9,i (s) analytic in Re s > 0, and the result is plain.
Proof of Theorem 8.7
Part (i) Define d to be the least common multiple of the di and (for a given
positive real ν) set N = d[ν/d]. Then ν P and N P differ by at most a finite
collection of lines of the form li : ai x − bi y = ci n, ci = 0, with N ≤ n ≤ ν.
8.8 Proofs
281
Hence Aν (P, Q, s) differs from A∗N (P, Q, s) by a finite sum of 12 An (li ∩ P, Q, s)
with ci = 0 and N ≤ n ≤ ν, which is exactly 12 A N (li ∩ P, Q, s) for any radial
lines (i.e., lines with ci = 0), together with a finite set of points lying at the
intersections of these various lines.
However, we have already seen (see (8.8.4)) that when ai /di and bi /di are of
opposite parity the lines ai x + bi y = nci , ci = 0, contribute O((ci n/di )−2σ ) =
O(ν −2σ ) while, for the radial lines, we found (see (8.8.5)) a contribution Ci (s) +
O(N −2σ ), with Ci (s) analytic for Re s > 0 and Ci (s) = 0 unless (0, 0) lies
on li ∩ P. The left-over points, being of distance ν from the origin, similarly
contribute only terms of size O(ν −2σ ), and so the limit exists for all Re s > 0.
Part (ii) Given our polygon with sides li : ai x − bi y = ci we set
δ=
1
2
min |ci |−1
i
and observe that, for an integer N , in going from N P to (N + δ)P we may lose
some lines of points but can gain no new lattice points and, similarly, in going
from (N − δ)P to N P we may gain but cannot lose lattice points. Hence the two
differences
|A N ±δ (P, Q, δ) − A N (P, Q, δ)|
consist solely of sums of A N (li ∩ P, Q, σ ) with ci = 0 (together with odd points
of intersection that are of size O(N −2σ )). Further, if we take N = 2nd (where d
is the least common multiple of all the gcd(ai , bi )) then every li with ci = 0 will
appear in one of these sums (which sum will depend on whether (0, 0) lies on the
same or the opposite side of the line as does the interior of the polygon). As in
part (i) the lines li with ai /di and bi /di of opposite parity and ci = 0 will only
contribute O(N −2σ ). However, if a I /d I and b I /d I are both odd and c I = 0 then
A N (l I ∩ T, Q, σ ) = (−1)ci (2nd)/d I
f n (t)
κ1 N ≤t≤κ2 N
will contribute
κ1 N ≤t≤κ2 N
f n (t) ≥ e1
N −2σ ≥ e2 N 1−2α ,
κ1 N ≤t≤κ2 N
for some positive constants ei . Hence, if 0 < σ ≤ 12 and we have at least one l I
with a I /d I and b I /d I both odd and c I = 0 then at least one of |A N ±δ (P, Q, δ) −
A N (P, Q, δ)| will always be bounded away from 0 by a constant (irrespective of
282
Madelung sums in higher dimensions
N ) and the limit A(P, Q, σ ) cannot exist. When P is a star body centred at (0,0)
we set δ = 1 and (since we only gain points in going from (N − 1)P to P) the
same argument shows that A N (P, Q, σ ) − A N −1 (P, Q, σ ) does not tend to zero
with N ; hence the limit does not exist even if we restrict ourselves (as is natural)
just to integer scalings of P. When all the li with ci = 0 have ai /di , bi /di of
opposite parity but there are radial lines (l1 through lk say) with ci = 0 and ai /di ,
bi /di both odd we consider A N (P, Q, σ ) for N = nd. By the theorem it is clear
that convergence for 0 < σ ≤ 12 will be determined solely by the sum over
the perimeter. As in part (i) the lines li , i > k, cannot disturb the convergence.
However, the radial lines li , i = 1, . . . , k, each contribute
A N (li ∩ P, Q, σ ) =
f n (t) = f (Pi )
t −2σ
κ1 N ≤t≤κ2 N
log N
N 1−2σ
≥ e3
t =0
κ1 N ≤t≤κ2 N
if σ = 12 ,
if σ < 12
(where e3 is some positive constant) and the resulting sum is plainly unbounded
as N → ∞.
The proof of Corollary 8.3 is immediate from Theorem 8.7.
Proof of Theorem 8.8 We split P into a series of cones Pi each of whose base is
a face of P and each with vertex (0, 0, 0):
Pi := {(x, y, z) : ai x + bi y + ci z ≤ ei , αi j x + βi j y + γi j z ≤ 0, 1 ≤ j ≤ Ji },
for some integers αi j , βi j , γi j , so that
A N (P, Q, s) =
I
A!N (Pi , Q, s),
i=1
A!N (P,
where
Q, s) indicates that points on the sides of N P (excepting the base)
are to be counted with weight 12 . Slicing up each three-dimensional polygon N Pi
into two-dimensional polygons Pi,m parallel to its base, where
Pi,m :={(x, y, z) ∈ Z2 : ai x + bi y + ci z
=m, αi j x + βi j y + γi j z ≤ 0, 1 ≤ j ≤ Ji },
we have
A!N (Pi , Q, s) =
N ei ∗
m=1 x∈Pi,m
di |m
Q(
x )−s ,
where ∗ denotes that points on the boundary of the polygon are to be counted
with weight 12 .
We suppose now that Pi,m comes from a face with at least one of ai /di , bi /di , or
ci /di even. We assume (replacing m by di m as necessary) that gcd(ai , bi , ci ) = 1
8.8 Proofs
283
(reordering as necessary), that ci is odd, and that ai is even. Setting αi =
gcd(ai , ci ) and choosing an even integer bi such that bi bi ≡ 1 (mod αi ) and an
odd integer (ai /αi ) such that (ai /αi )(ai /αi ) ≡ 1 (mod ci /αi ) we make the change
of variables demanded by the relation ai x + bi y ≡ m (mod ci ) on Pi,m , i.e.,
y = bi m + αi y and
x=
ai
αi
1 − bi bi
αi
m−
ai
ci
bi y +
x .
αi
αi
Observe that ai x + bi y + ci z = m becomes
1 − (ai /αi ) (ai /αi )
1 − bi bi
ai
−
x
z =m
αi
ci /αi
αi
1 − (ai /αi ) (ai /αi ) y
− bi
ci /αi
and that
(−1)x+y+z = (−1)x +y .
Hence the sum of (x, y, z) over Pi,m is replaced by a sum of (x , y ) over
m Ri ∩ Z2 , where Ri is the polygon
Ri = {(x , y ) ∈ Z2 : αi j x + βi j y ≤ γij , 1 ≤ j ≤ Ji }
and the αi j , βi j , γij are integers with, we shall assume, no common factor; we
define di j :=gcd(αi j , βi j ). Writing
Q(x, y, z) = Q i (x , y , m)
(where Q i (x, y, z) is a positive definite quadratic form), we have
∗
∗
Q(x, y, z)−s =
(−1)x+y Q i (x, y, m)−s =: A∗m (Ri , Q i , s),
(x,y,z)∈Pi,m
(x,y)∈m Ri ∩Z2
here ∗ denotes that points on the perimeter are counted with weight 12 . To
evaluate the A∗m (Ri , Q i , s) one proceeds almost exactly as in the proof of
Proposition 8.1 (replacing the bounds for the kth derivative, k = 0, 1, 2 in
(8.8.3) by
| f n(k) (t)| = O (|n| + |m|)−2σ −k
and so on), to give
†
(−1)x+y Q i (x, y, m)−s = O(m −2σ ),
(x,y)∈m Ri ∩Z2
284
Madelung sums in higher dimensions
where † denotes that for each side li j of the polygon Ri the last line of points
parallel to that side,
Li j (m) := m
γij
Ri j ,
di j
Ri j :=
di j
(li j ∩ Ri )
γij
is to be included in the sum with weight 12 . The desired sum A∗ (Ri , Q i , s) thus
differs from this latter sum only by the addition or exclusion of the (half-weighted)
expressions for those sides li j for which m ≡ 0 (mod di j ); furthermore,
A∗m (Ri ,
Q i , s) = O(m
−2σ
Ji
1
)+
Ai j (m),
2
j=1
di j|m
where
Ai j (m) := ±
(−1)x+y Q 1 (x, y, m)−s ,
(x,y)∈Li j (m)
the ± sign being determined by whether (0,0) lies respectively on the interior or
exterior side of li j . Thus
A!N (Pi , Q, s) = Ci,0 (s) +
Ji
E i j + O(N 1−2σ ),
j=1
with
E i j :=
Ai j (m)
1≤m≤N ei /di
di j|m
and Ci,0 (s) analytic for all Re s > 12 .
Now, if αij := αi j /di j and βij := βi j /di j are of opposite parity then one
readily shows that alternation in sign along the line Li j (m) gives (in the manner
of (8.8.4))
Ai j (m) = O(m −2σ ),
and hence
E i j = Ci j (s) + O(N 1−2σ )
with the Ci j (s) analytic for Re s > 12 .
If the αij , βij are both odd, parameterizing the line as in the proof of
Proposition 8.1 we have
(−1)x+y Q i (x, y, m)−s
An (Ri j , Q i (m), s) :=
(x,y)∈n Ri j ∩Z2
= (−1)n
nξ1 ≤t≤nξ2
Q 1 (nx0 − αij , βij t − ny0 , m)−s ,
8.8 Proofs
285
for some fixed integers x0 = x0 (i, j), y0 = y0 (i, j), with αij x0 − βij y0 = 1, and
rational numbers ξk = ξk (i, j). So, for bounded integers k1 and k2 we certainly
have
An+k1 (Ri j , Q i (m + k2 ), s) = (−1)k1 An (Ri j , Q i (m), s) + O (|m| + |n|)−2σ ,
where trivial bounding gives
An (Ri j , Q 1 (m), s) = O
n
(|n| + |m|)2σ
.
Hence, splitting the sum over m into multiples of 2di j we have
Ai j (2di j l) +
O(m −2σ ) + O(N 1−2σ ),
E i j = Si j
l≤N ei /2di di j
1≤m≤N ei /di
where, as may be readily checked,
Si j :=
2di j
(−1)
mγij /di j
= 0,
m=1
di j |m
giving
E i j = Ci j (s) + O(N 1−2σ )
with Ci j (s) analytic for Re s > 12 .
Therefore
A!N (Pi , Q, s) = Ci (s) + O(N 1−2σ ),
with Ci (s) analytic for all Re s > 12 , and part (i) of the theorem follows at once if
all the faces have at least one of ai /di , bi /di , ci /di even.
For part (ii) one proceeds in the manner of the proof of Theorem 8.2 to show,
by counting the number of points on the faces with ai /di , bi /di , ci /di all odd, that
(for suitable multiples N and δ := 12 mini ei−1 ) the contribution from those faces
to |A N (P, Q, s) − A N −δ (P, Q, s)| does not tend to zero as N → ∞.
Proof of Theorem 8.9 Our approach resembles the proof of Theorems 5 and 6 in
Borwein et al. [4, Section IV]. We set
T (z) :=
Tm (z) :=
∞
n 1 =−∞
m1
n 1 =−m 1
∞
···
q(n 1 , . . . , n k )z Q(n 1 ,...,n k ) ,
n k =−∞
mk
···
q(n 1 , . . . , n k )z Q(n 1 ,...,n k ) ,
n k =−m k
and define the normalized Mellin transform Ms [ f ] for Re s > 0 by
∞
Ms [ f ] := −1 (s)
f (t)t s−1 dt.
0
286
Madelung sums in higher dimensions
We set
F(s) := Ms [T (e−t ) − q(0, . . . , 0)]
and observe that (since Ms [e−at ] = a −s for a > 0)
Am (s) = Ms [Tm (e−t ) − q(0, . . . , 0)].
We shall need the following uniform-boundedness lemma.
Lemma 8.3 For any t > 0 and integers Ni ≥ 0 (with at least one Ni > 0)
∞
∞
−t Q(n 1 ,...,n k ) −λ(N12 +···+Nk2 )t
S := ···
q(n 1 , . . . , n k )e
< Ce
n 1 =N1
n k =Nk
for some C = C(Q) > 0 and λ = λ(Q) > 0.
Proof We set
g(u 1 , . . . , u k ) := e−Q(u 1 ,...,u k )
and write χ N for the characteristic function of the region
{(u 1 , . . . , u k ) ∈ Rk : u i ≥ Ni }.
Applying the partial summation technique employed in the proof of Theorem 8.1,
∞
∞ n
a(n)b(n) =
n=−∞
a(l) [b(n) − b(n + 1)]
n=−∞ l=−∞
∞ n
=−
n=−∞
l=−∞
n+1
a(l)
b (u) du,
n
to each of the variables n i in turn we obtain
∞
∞
···
W (n 1 , . . . , n k )I (n 1 , . . . , n k ) ,
S=
n 1 =−∞
n k =−∞
where
n1
W (n 1 , . . . , n k ) :=
···
r1 =−∞
nk
rk =−∞
q(r1 , . . . , rk )χ N (r1 , . . . , rk )
and
I (n 1 , . . . , n k ) :=
n 1 +1
n1
···
n k +1
nk
√
√
∂
∂
···
g(u 1 t, . . . , u k t) du 1 · · · du k .
∂u 1
∂u k
8.8 Proofs
287
By assumption, |W (n 1 , . . . , n k )| < B (which vanishes unless n i ≥ Ni ), hence
∞
∞
∂
√
√ ∂
S≤B
···
···
g(u 1 t, . . . , u k t) du 1 · · · du k
∂u k
N1
Nk ∂u 1
∞
∞ ∂
∂
=B
···
g(u 1 , . . . , u k ) du 1 · · · du k .
√ ···
√ ∂u k
N1 t
Nk t ∂u 1
Now, since Q(x1 , . . . , xk ) is positive definite we have
Q(x1 , . . . , xk ) > 2λ(x12 + · · · + xk2 )
for some λ > 0, giving
∂
∂
−2λ(u 21 +···+u 2k )
∂u · · · ∂u g(u 1 , . . . , u k ) ≤ |P(u 1 , . . . , u k )|e
1
k
2
2
= O e−λ(u 1 +···+u k ) ,
where P(u 1 , . . . , u k ) = PQ (u 1 , . . . , u k ) is some polynomial of total degree k.
Thus
∞
∞
2
2
−λ(u 21 +···+u 2k )
du 1 · · · du k = O e−λ(N1 +···+Nk )t ,
S=
√ ···
√ O e
N1 t
Nk t
as claimed.
Observing that (replacing u i by −u i as necessary) T (e−t ) − q(0, . . . , 0) can
be written as a sum of sums of the form S with at least one Ni ≥ 1 and that
T (e−t ) − Tm (e−λt ) can be written as a sum of sums with at least one Ni ≥ m =
min m i , we have
&
%
2
T (e−t ) − Tm̄ (e−t ) = O e−λm t .
T (e−t ) − q(0, . . . , 0) = O e−λt ,
Hence
∞
F(s) = −1 (s)
T (e−t ) − q(0, . . . , 0) t s−1 dt
0
∞
(σ )
= O −1 (s)
e−λt t σ −1 dt = O
|(s)|
0
exists, and similarly
−1 |F(s) − Am (s)| = O (s)
0
∞
e
−λm 2 t σ −1
t
dt
−2σ (σ )
→0
=O m
|(s)|
as m → ∞ in any region {s : σ > ε, |s| ≤ K } for a fixed positive ε and K .
Thus the limit exists and is analytic for all Re s > 0.
288
Madelung sums in higher dimensions
Proof of the Section 8.5 formulae We recall the theta functions
∞
θ2 (q) :=
q
(n+1/2)2
,
∞
θ3 (q) :=
n=−∞
qn
2
n=−∞
and observe (as may be deduced from Zucker and Robertson [24, 25]) that when
√
Q( − p) is a unique factorization domain containing u units we have
∞
r (n, Q p )q := θ2 (q)θ2 (q ) + θ3 (q)θ3 (q ) = 1 + u
n
p
p
n=1
∞
(− p|n)
n=1
qn
.
(1 − q n )
Hence r (n, Q p ), the number of integer representations of n by Q p (x, y), satisfies
r (n, Q p ) = u
(− p | d),
d|n
and (8.4.2) is plain.
When p = 3 or 7 we can relate r (n, p), the number of integer solutions of
√
2
x + py 2 = n, to r (n, Q p ). Setting N (x + y − p) = x 2 + py 2 and writing
Q p (x, y) = 14 (x − y)2 + 14 p(x + y)2 ,
it is easily seen that r (n, p) represents the number of integer solutions (x, y) of
√
N (x + y − p) = n and r (n, Q p ) represents the number of integer solutions
√
(X, Y ) of N ((X/2) + (Y/2) − p) = n, with X and Y of the same parity. It is
easily seen (matching (x, y) to (X, Y )) that r (4n, p) = r (n, Q p ) and, by congruences modulo 4, that r (n, p) = 0 when n ≡ 2 (mod 4). When n is odd and
√
p = 7, congruences modulo 8 show that if N ((X/2) + (Y/2) − p) = n then
X and Y are necessarily even (giving a pairing of (x, y) and (X/2,
%
& and
√Y/2))
r (n, 7) = r (n, Q 7 ). When n is odd and p = 3, by putting ω = 1 + −3√ /2 it
solutions ((X/2)+(Y/2) −3),
is readily checked√that exactly one of the r (n, Q 3 )√
((X/2) + (Y/2) −3)ω, and ((X/2) + (Y/2) −3)ω2 will be of the form
√
(x + y − p), and hence that r (n, 3) = r (n, Q 3 )/3.
Thus, setting λ = 1 or 1/3 as p = 7 or 3 respectively, we have
∞
(±1)n r (n, p)
n=1
ns
=
∞
r (n, Q p )
n=1
(4n)s
±λ
∞
r (n, Q p )
ns
n=1
n odd
= 4−s ± λ(1 − 2−s )(1 − (− p|2)2−s )
∞
r (n, Q p )
n=1
ns
.
8.9 Commentary: Alternating series test
289
So, finally, noting that (−3 | 2) = −1, (−7 | 2) = 1, and (1 − 21−s )ζ (s) = α(s),
we have
1
= 2(1 + 21−2s )ζ (s)L −3 (s),
2 + 3y 2 )s
(x
2
(x,y)∈Z \(0,0)
(x,y)∈Z2 \(0,0)
(x,y)∈Z2 \(0,0)
(x,y)∈Z2 \(0,0)
(−1)x+y
= −2(1 + 21−s )α(s)L −3 (s),
(x 2 + 3y 2 )s
1
= 2(1 − 21−s + 21−2s )ζ (s)L −7 (s),
(x 2 + 7y 2 )s
(−1)x+y
= −2α(s)L −7 (s).
(x 2 + 7y 2 )s
8.9 Commentary: Alternating series test
A mapping ā : N N → R is (N -)monotone if, for m 1 , m 2 , . . . , m N ≥ 0,
N (−1)
si
ā(m 1 + s1 , m 2 + s2 , . . . , m N + s N ) ≥ 0.
i=1 s1 =0,1
Thus 1-monotonicity corresponds to ā(m) ≥ ā(m + 1), and 2-monotonicity
corresponds to
ā(m, n) + ā(m + 1, n + 1) ≥ ā(m, n + 1) + ā(m + 1, n),
while 3-monotonicity demands that the alternating sum over any unit cube be
positive if the bottom corner is positive. We say that ā is fully monotone if ā and
all its restrictions are monotone. Consider
N ∞
(−1)
mi
ā(m 1 , m 2 , . . . , m N ).
i=1 m i =0
(1) As was done in [2], one may prove the following. If ā is fully monotone and
limn̄→∞ ā(m̄) = 0 then the rectangular sums converge in an alternating fashion.
For instance, in two dimensions one needs to show that
sn :=
n
(−1)i+ j ā(i, j)
i, j=0
satisfies s2n ≥ s2n+2 , s2n+1 ≥ s2n−1 , and s2n − s2n−1 → 0. Drawing a picture
helps. Details for two and three dimensions are spelled out in [4, Chapter 2].
(2) Suppose that ā is N times continuously differentiable on (R+ ) N . One may
show that ā is totally monotone if the partial derivatives satisfy
āi1 ≤ 0,
āi1 i2 ≥ 0,
āi1 i2 i3 ≤ 0,
...,
(−1) N āi1 i2 ···i N ≥ 0
290
Madelung sums in higher dimensions
for all partial derivatives with i 1 < i 2 < i 3 < . . . < i N .
i+ j+k (i 2 + j 2 + k 2 )− p and
Hence, one can check that sums such as ∞
0 (−1)
∞
i+
j
−
p
(2i + j) converge for p > 0.
0 (−1)
8.10 Commentary: Hurwitz zeta function
The notation (k, l : s) introduced in Section 4.2 (see (4.2.5)) can be also expressed
in terms of the Hurwitz zeta function ζ (s, q). Here we summarize a few basic
properties of ζ (s, q).
(1) First note that ζ (s, 1) = ζ (s) and ζ (s, 12 ) = (2s − 1)ζ (s), and we have a
multiplication theorem:
k ζ (s) =
s
k
ζ (s, m/k).
m=1
By using a Mellin transform, we obtain the following integral representation for
ζ (s, q), valid when Re s > 1 and Re q > 0:
∞ s−1 −qt
t e
1
dt,
(8.10.1)
ζ (s, q) =
(s) 0 1 − e−t
Equation (8.10.1) can be used to provide an analytic continuation for ζ (s, q) to
the whole complex plane except for a simple pole at s = 1. In particular, when s
is a non-positive integer we have ζ (−n, q) = −Bn+1 (q)/(n + 1), where Bn (x)
denotes the nth Bernoulli polynomial, while when s is a positive integer, ζ (s, q)
may be expressed in terms of the polygamma function. Many other special values
are known, see for instance [1]; [19] is also a useful reference.
When aided by a computer algebra system, (8.10.1) facilitates the rapid evaluation of ζ (s, q), and hence of real and complex Dirichlet L-series, to high
precision. In particular, if we make the change of variable x k = e−t in (8.10.1)
then
(−1)s−1 1 x l−1 (log x)s−1
d x,
(8.10.2)
(k, l : s) =
(s)
1 − xk
0
which is conducive to both numerical and symbolic computations. For example,
suppose that we require to evaluate L i−5 (1) in (4.4.9) at s = 1. Using (8.10.2), we
have
1
1 + (1 + i)x + x 2
i
d x.
L −5 (1) =
2
3
4
0 1+x +x +x +x
√
√
&%
&
%
The denominator of the integrand factorizes as x 2 + 1+2 5 x +1 x 2 + 1−2 5 x +1 ,
so we can split the integrand into partial fractions, and the evaluation follows.
For some other L-series the resulting integral may be evaluated using the residue
theorem. In all cases, computer algebra systems are a significant aid.
8.10 Commentary: Hurwitz zeta function
291
(2) The n-dimensional Hurwitz zeta function is described in [5]. Let d > 0, ā :=
(a1 , a2 , . . . , a N ), si := sign(ai ), and define a function
ni
∞
si
L ā (s, d) :=
.
(8.10.3)
( |ai |n i + d)s
{i|ai =0} n i =0
n
−s
So, for instance, L −1,2,3 (s, 1) = ∞
n,m, p=0 (−1) (n + 2m + 3 p + 1) .
Using a Mellin transform (much as we did for the one-dimensional Hurwitz
zeta function), we can show that
1 d−1
x
(− log x)s−1
1
d x.
(8.10.4)
L ā (s, d) =
ai
(s) 0
ai =0 (1 − si x )
N (so
If we let A±N (s, d) :=
±ē (s, d), where ē = (1, 1, . . . , 1) ∈ R
$ 1 Ld−1
−1
s−1
−N
(− log x) (1 ∓ x) d x), then via a Mellin
that A±N (s, d) = (s)
0 x
transform we have
A±N (s, d) =
∞
(m + N − 1)! (±1)m
1
,
(N − 1)!
m!
(m + d)s
(8.10.5)
m=0
which reduces as a sum of one-dimensional Hurwitz zeta functions after one
expands the ratio of factorials.
Therefore, using partial fractions on (8.10.4) and also using equation (8.10.5),
it transpires that all n-dimensional Hurwitz zeta functions factor into a linear
combination of one-dimensional Hurwitz zeta functions.
In many cases, we may compute A±N (s, d) recursively. Using integration by
parts, we have
∓1
(d − 1)A±(N −1) (s, d − 1) − A±(N −1) (s − 1, d − 1) .
N −1
(8.10.6)
When d ≥ N , the recursion (8.10.6) reduces A±N to a linear combination of onedimensional Hurwitz zeta functions. When d < N and d ∈ N, the recursion stops
at terms of the form A±N (s, 1), which by (8.10.5) evaluates to
N [N , i]
ζ (s − i),
A N +1 (s, 1) =
N!
A±N (s, d) =
i=0
and A−(N +1) (s, 1) has an identical expression except that an extra factor 1 −
21+i−s is inserted into the summand; here [n, k] denotes the (signed) Stirling
number of the first kind, which satisfies the generating function
n
[n, k] x k = (x)n = x(x − 1) · · · (x − n + 1)
k=0
and the recurrence [n, k] = (1 − n) [n − 1, k] + [n − 1, k − 1].
292
Madelung sums in higher dimensions
(3) We record some specific evaluations below; some can be computed in more
elementary ways than using the full procedure described above.
Probably the easiest example is
∞
m,n=1
1
= ζ (s − 1) − ζ (s);
(m + n)s
this can be readily seen by the change of variable m + n = k. Some related sums
are n,m (|m| + |n|)−s = 4ζ (s − 1), m,n>0 (−1)m+1 (m + n)−s = ζ (s)/2s , etc.
More generally,
n i >0
(m − 1)!
=
[m, i] ζ (s − i + 1).
s
(n 1 + n 2 + · · · + n m )
m
i=1
Other examples include
m,n≥0
k,m,n>0
(−1)m+n
(1 − 2s )(1 − 2s−1 )
L −4 (s)
=
ζ (s) +
,
s
2s
(2m + n + 1)
2
2
3(1 − 22−s )
(−1)k+m+n+1
1 − 23−s
ζ (s − 2) −
ζ (s − 1)
=
s
(k + m + n)
2
2
+ (1 − 21−s )ζ (s),
and
√
(−1)m+n
2 log 2
1
3π
=
−
− ,
3n + m
3
27
6
m,n>0
(−1)m+n
5π 2 − 27L −3 (2) − 36 log 2
,
=
2
108
(3n + m)
m,n>0
where we need to resort to integrals for the last two sums. The integral (8.10.4)
also provides identities such as
∞
a1 ,...,an =1
(−1)a1 +···+an +n
= (2n , 2n − 1 : s) − 2−ns ζ (s).
(a1 + 2a2 + 4a3 + · · · + 2n−1 an )s
References
[1] V. Adamchik. On the Hurwitz function for rational arguments. Appl. Math. Comp.,
187:3–12, 2007.
[2] D. Borwein and J. M. Borwein. A note on alternating series in several dimensions.
Amer. Math. Mon., 93:531–539, 1985.
[3] D. Borwein, J. M. Borwein, and R. Shail. Analysis of certain lattice sums. J. Math.
Anal. Appl., 143:126–137, 1989.
References
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[4] D. Borwein, J. M. Borwein, and K. Taylor. Convergence of lattice sums and
Madelung’s constant. J. Math. Phys., 26:2999–3009, 1985.
[5] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number
Theory and Computational Complexity. Wiley, New York, 1987.
[6] J. P. Buhler and R. E. Crandall. On the convergence problem for lattice sums. J. Phys.
A: Math. Gen., 23:2523–2528, 1990.
[7] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and
Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980.
[8] G. H. Hardy and M. Riesz. The General Theory of Dirichlet Series. Cambridge Tracts
in Mathematics and Mathematical Physics, Cambridge University Press, Cambridge,
1915.
[9] M. N. Huxley. Exponential sums and lattice points II. Proc. London Math. Soc.,
66:279–301, 1993.
[10] E. Krätzel. Bemerkungen zu einem Gitterpunktsproblem. Math. Ann., 179:90–96,
1969.
[11] E. Krätzel and W. Nowak. Lattice points in large convex bodies, II. Acta Arith.,
62:285–295, 1992.
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Gitterpunkte in mehrdimensionalen Ellipsoiden). S. B. Preuss. Akad. Wiss., 458–476,
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[13] E. Landau. Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Math. Zeit.,
21:126–132, 1924.
[14] E. Landau. Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Zweite Abhandlung. Math. Zeit., 24:299–310, 1926.
[15] E. Landau. Vorselungen über Zahlentheorie, vol. 2., Part 8, Chapter 6. Chelsea, New
York, 1955.
[16] B. Novák. Über eine Methode der -abschätzungen. Czech. Math. J., 21:257–279,
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1976.
[18] M. Riesz. Sur un théorème de la moyenne et ses applications. Acta Univ. Hungaricae
Franc.-Jos., 1:114–126, 1923.
[19] H. M. Srivastava and J. Choi. Series Associated with the Zeta and Related Functions.
Kluwer Academic, Dordrecht, 2001.
[20] A. Walfisz. Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Math. Zeit.,
19:300–307, 1924.
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SSR. Trudy Tbiliss. Mat. Inst. Razmadze, 22:33–75, 1956.
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9
Seventy years of the Watson integrals
Watson [59] published in 1939 the evaluation of three integrals submitted to him,
which had arisen from a problem in physics [57]. Over the years these integrals
have continued to occur in other aspects of physics such as random walk problems. This article reviews these integrals and their generalizations over the past
70 years.
9.1 Introduction
Here we give a short history of three triple integrals and their generalizations;
the integrals first appeared some 70 years ago. This provides an example of how
physicists and mathematicians like to elaborate particular results and extend them
to more general areas. In doing so new techniques are developed, which eventually
become standard methods. It is also an example of how chance plays a part in
bringing together researchers with knowledge in different fields who collaborate
to produce results which might not otherwise have been obtained.
This will be a rather personal account, as many hundreds of papers and articles
concerning these integrals have been published, and it is not possible in limited
space to refer to every contribution, so at the start we apologize to authors whose
work we do not cite.
The three original integrals are
π π π
d x d y dz
1
,
WB = 3
π 0 0 0 1 − cos x cos y cos z
π π π
d x d y dz
1
WF = 3
,
π 0 0 0 3 − cos x cos y − cos y cos z − cos z cos x
π π π
d x d y dz
1
WS = 3
.
π 0 0 0 3 − cos x − cos y − cos z
(9.1.1)
(9.1.2)
(9.1.3)
9.1 Introduction
295
These integrals made their first appearance in a paper on magnetic anisotropy by
van Pepye [57] in 1938. They emerged in connection with three well-known cubic
structures formed by real crystals, namely the body-centred (B), face-centred (F),
and simple cubic (S) lattices. Van Pepye was a student of the Dutch physicist
H. A. Kramers, and clearly these integrals intrigued the latter. He sent them on to
R. H. Fowler, the son-in-law of Rutherford, in Cambridge. Let Watson [59], who
worked at Birmingham University, describe in his own words how the problem
reached him:
The problem of evaluating them was proposed by Kramers to R. H. Fowler who communicated them to G. H. Hardy. The problem then became common knowledge first
in Cambridge and subsequently in Oxford, whence it made the journey to Birmingham
without difficulty.
Whatever his motive for this last somewhat barbed comment, Watson found
closed forms for these integrals and they are now universally known by his name.
They will be referred to as WIs. The first, as Watson acknowledged, is fairly well
known. Indeed, van Pepye himself had evaluated it and the result can be traced
back to Kummer [40]; also, its generalization is simple to find (Maradudin et al.
[42]), so the closed form for this more general version of (9.1.1) can now be given.
We have
π π π
d x d y dz
1
W B (wb ) = 3
π 0 0 0 wb − cos x cos y cos z
(
'
1 1
4 2
(9.1.4)
−
1 − wb−2 .
= 2K
2 2
π
When wb = 1,
W B (1) =
1 4 1
4 2 1 K √ =
4 = 1.3932039297 . . . ,
2
2
π
4π 3
(9.1.5)
where K is the complete elliptic integral of the first kind.
As complete elliptic integrals of the first kind play a primary role in all aspects
of WI and their generalizations, it is apposite here to give a short account of them.
The complete elliptic integral of the first kind is defined by
1
π/2
dt
dx
=
(9.1.6)
K (k) :=
2
1/2
(1 − k 2 sin t)1/2
0
0 (1 − x 2 )(1 − k 2 x 2 )
for 0 ≤ k < 1, where k is known as the modulus. The complementary modulus
k is defined by the relation k 2 + k 2 = 1 and we write K (k ) := K (k). The
integrals K and i K play roles in the theory of the Jacobian elliptic functions
that are equivalent to that played by π/2 in the theory of the circular functions;
296
Seventy years of the Watson integrals
namely, they are quarter periods. An important property of K (k) is that it can be
expressed as a hypergeometric function:
∞ 1 1
(2n)! 2 2n
,2 2
π
π
2
k ,
(9.1.7)
K (k) = 2 F1
;k =
2
1
2
22n (n!)2
n=0
which will later be seen as crucial in our further analysis.
Although (9.1.1) and its generalization were easily solved, (9.1.2) and (9.1.3)
are a different matter. Watson’s approach [59] was to use an inspired sequence
of changes of variable and integrations to reduce the triple integrals to single
integrals involving K . Then, using expansions of K and K that he himself had
obtained 30 years previously [58] he was able to evaluate these last integrals. One
should go to the original paper to admire the ingenuity displayed in finding (9.1.8)
and to enjoy the brilliance of his derivation of (9.1.9). His results are
√
√
6 1
3
3
3
3−1
(9.1.8)
= 14/3 4 = 0.4482203944 . . . ,
WF = 2 K 2
√
π
2
π
2 2
√
√
√
√ √ √
4(18 + 12 2 − 10 3 − 7 6) 2 (2
−
WS =
K
3)(
3
−
2)
π2
= 0.5054620197 . . . .
(9.1.9)
It will be observed that whereas W B and W F are expressed in terms of gamma
functions, this was not done for W S and had to await a later investigation.
The next appearance of WIs in the literature (though not by name) followed
almost immediately. McCrea and Whipple, in an odd paper [44], took up the problem of the ‘drunkard’s walk’ in three dimensions. The strangeness of the paper
was not in its content but in the complete obliviousness of the authors to any
other work in this field. There were only two references given. The first was to a
note of McCrea in the Mathematical Gazette [43], in which he had considered the
two-dimensional square lattice and, as a kind of afterthought, the second was to a
paper by Courant et al. [13], which had been brought to their attention. The problem may be formulated as follows. An intoxicated person drops his house keys
under a lamppost – the origin. Suppose the lamppost is one of an infinite number of posts equally spaced along a line. The inebriate staggers from lamppost to
lamppost with an equal probability of going to the right or left while looking for
the keys. What is the probability that he or she will return to the origin? The same
question may be asked if the lampposts form a two-dimensional square array or
a three-dimensional cubic array. In fact this problem was first discussed by Pòlya
[48]. He showed that in the one- and two-dimensional cases the probability of
return to the origin was unity, i.e., the keys would eventually be found. However,
in three dimensions the probability was less than one, but Pòlya did not find a
value for it.
9.1 Introduction
297
This problem is mirrored in one dimension exactly by the tossing of a coin
(heads or tails), in two dimensions by the throwing of a tetrahedral die, and in
three dimensions by the casting of a common six-sided die. Stewart [55] discusses
this in some detail. He points out that in tossing a fair coin the a priori probability is equal numbers of heads and tails. As one continues tossing, an imbalance
between the numbers of heads and tails will occur but, no matter how large this
imbalance becomes, if you carry on flipping the coin the imbalance will eventually correct itself. Similarly, in throwing an unbiased tetrahedral die, the digits 1,
2, 3, and 4 have an equal probability, of 1/4 of appearing. As before one starts
with equal numbers of the four digits and again, as one continues throwing, deviations from the initial state will appear. But, as in the coin case, if one continues to
throw the four-sided die, the state of equal numbers for the four digits will eventually reoccur. However, in throwing an ordinary six-sided die, each of the digits
1–6 has a probability of 1/6 of occurring. Starting from equal numbers of the six
digits, imbalances will develop, but the probability of returning to equal numbers
of the six digits is less than unity. Stewart actually credits Ulam as proving this,
but gives no reference, and we think that Pòlya [48] must be credited as being first
with this result.
However, it was McCrea and Whipple who first set out to find a value for this
probability. Clearly they did not know of Pòlya’s paper, for no reference to him
is made, but they succeeded in showing that the probability of return to the origin
was 1 − 1/(3W S )! (We have converted their result into the notation used here. All
authors in this field use their own symbols, so all formulae given here are translations of their results into notation consistent with that used here.) Just as McCrea
and Whipple were unaware of Pòlya they also were in ignorance of Watson, and
it is interesting to follow their attempt to evaluate 3W S . They rapidly reduced W S
to a single integral from π/6 to π/2 of a complete elliptic integral multiplied by
an angular factor and then resorted to numerical integration. Such was the inaccuracy of their computation that they only gave the value to two significant figures,
3W S = 1.53, which, compared with the value 1.51638606 found from (9.1.9), is
woefully in error. They thus found the probability of returning to the origin was
0.34 – surprisingly small. This seems to be the very first numerical evaluation of
this probability.
A much more accurate result was found by Domb [18] in a completely different
manner with no reference to W S whatsoever. He calculated directly the probability
of the return to the origin, pn , after n steps. For n odd you cannot return to your
starting point and thus p2n+1 = 0. For small n, p2n can be calculated exactly
from a rather complex formula. In this way he directly calculated p2 , . . . , p18 ;
then, applying Euler–Maclaurin summation to the asymptotic formula for p2n , he
obtained an excellent estimate 1.51639, for the equivalent of 3W S , giving 0.34054
for the probability of return to the origin. If we use the more accurate value of
Watson, the probability of return to the origin is 0.340537330.
298
Seventy years of the Watson integrals
Now, the Courant paper referred to by McCrea and Whipple was actually a
seminal paper on how limiting forms of difference equations could equal the solution of the equivalent differential equation. It was noted that the random walk
problem was set up as a difference equation and the solution was none other than
the Green’s function of the equation u = 0. This idea was elaborated on by Duffin [20], who considered an infinite lattice in which every point was connected to
its nearest neighbour by a unit resistance. If a current is introduced at some lattice
point then it will split equally among all connections, and so on at every lattice
point, and this again mimics the random walk problem. Duffin then considered an
infinite simple cubic lattice in which every lattice point is labelled (l, m, n); each
of l, m, and n may take on every integer value from −∞ to ∞. A unit current
is introduced into the source point (0, 0, 0). The Green’s function G(l, m, n) is
defined by the solution to
DG(0, 0, 0) = −1,
DG(l, m, n) = 0 otherwise,
where D is the difference operator corresponding to the Laplacian differential
operator in three dimensions. The solution is found to be
G(l, m, n) =
1
π3
π
0
π
0
π
0
cos lx cos my cos nz
d x d y dz.
3 − cos x − cos y − cos z
(9.1.10)
This result was ascribed to Courant [12]. A similar result was then obtained
by Davies [14] without reference to Duffin or Courant. Davies noted that the
resistance between the source point and a point infinitely far away was just
G(0, 0, 0); this is of course just the Watson integral (9.1.3). Also, the resistance between the source point and any other point (l, m, n) on the lattice was
Rl,m,n = G(0, 0, 0) − G(l, m, n). An excellent modern account of the relationship between random walks on lattices and resistance networks has been given by
Doyle and Snell [19].
The association of WIs as special values of Green’s functions of a particular
lattice has led to the term Green’s function being applied to simple generalizations
of WIs. Thus it has became customary to refer to any integral of the form
1
π3
π
0
π
0
π
0
d x d y dz
,
w − φ(cos x, cos y, cos z)
(9.1.11)
where φ(cos x, cos y, cos z) is some ternary trigonometric polynomial, as a lattice
Green’s function. However, it would be better to think about objects having the
form (9.1.11) as functions of a complex variable w. What Watson did was to evaluate the original integrals at a certain critical value of w relevant to each integral,
where the integrand becomes infinite at the origin. It thus became a challenge to
extend his analysis to general w, i.e., to evaluate
9.2 Solutions for W F (w f ), W F (α f , w f ), and W S (ws )
1
W F (w f ) = 3
π
W S (ws ) =
1
π3
π
0
π
0
π
0
0
π
0
π
0
π
299
d x d y dz
,
w f − cos x cos y − cos y cos z − cos z cos x
(9.1.12)
d x d y dz
.
(9.1.13)
ws − cos x − cos y − cos z
Then (9.1.12) defines a single-valued analytic function in the w f plane, provided
that a cut is made along the real axis −1 < w f ≤ 3. A similar property holds
for (9.1.13) provided that the cut on the real axis is made for −3 < ws ≤ 3.
Also, rather than having a different w for each lattice, many authors considered a
modified form of WI, namely
π π π
d x d y dz
1
,
(9.1.14)
P(u) = 3
π 0 0 0 1 − (u/d)φ(cos x, cos y, cos z)
where d is the number of terms in φ. Sometimes it is more convenient to deal with
P. It is the evaluation of such quantities that has exercised investigators from the
mid 1950s to present times.
The first extension of Watson’s result was achieved by Montroll [47]. He
considered the integral
π π π
d x d y dz
1
W S (2 + α, α) = 3
π 0 0 0 2 + α − α cos x − cos y − cos z
and, following exactly the same path as Watson, Montroll found
√
25/2 k1 k2 K (k1 )K (k2 )
W S (2 + α, α) =
,
(9.1.15)
√
απ 2
where
√
√
1 √
( 4 + 2α − 2)(2 1 + α − 4 + 2α) ,
k1 =
2α
√
√
1 √
k2 =
( 4 + 2α + 2)(2 1 + α − 4 + 2α) .
2α
Although this indicated that further generalizations might be possible, no new
methods were introduced and the chances of improving on Watson’s ingenuity
seemed small. Two big advances were made, however, when Iwata [29] produced
a closed-form solution for W F (w f ) and when Joyce [31, 32] further generalized
W F but more importantly found a closed form for W S (ws ).
9.2 Solutions for W F (w f ), W F (α f , w f ), and W S (ws )
Iwata’s method of finding a closed form for (9.1.12) was as follows. He copied
Watson in reducing the triple integral to a single integral, but did it in an
entirely different fashion. He made use of two successive well-known integrations, and then solved this last integral in such a way that this has become a
300
Seventy years of the Watson integrals
standard integration. The first integration was accomplished using the established
result
π
π
dz
= 2
,
(9.2.1)
a
−
b
cos
z
(a
−
b2 )1/2
0
which after integrating (9.1.12) over z gives
W F (w f ) =
1
π2
π
0
π
0
dx dy
.
[(w f − cos x cos y + cos x + cos y)(w f − cos x cos y − cos x − cos y)]1/2
Then substituting cos y = u, after some re-arrangement one obtains
π
dx
W F (w f ) =
2 x − 1)1/2
(cos
0
−1/2
1 w f + cos x
w f − cos x
2
−u
− u (1 − u )
×
du.
cos x − 1
cos x + 1
−1
(9.2.2)
The standard result
1
−1
du
2
=
K (k),
2
1/2
[(a − u)(b − u)(1 − u )]
[(a − 1)(b + 1)]1/2
(9.2.3)
where
k2 =
2(a − b)
,
(a − 1)(b + 1)
is now employed on (9.2.2) and after some algebra W F (w f ) is obtained as a single
integral, namely
⎡ '
⎤
π
2 w f + cos2 x
2
⎦ d x.
W F (w f ) = 2
K⎣
(9.2.4)
wf + 1
π (w f + 1) 0
There are a number of other ways in which this integral may be expressed; thus
for example
π √
2
K
A + B cos x d x,
(9.2.5)
W F (w f ) = 2
π (w f + 1) 0
4w f + 2
2
A=
,
B=
,
(w f + 1)2
(w f + 1)2
π 2
W F (w f ) = 2
K
a 2 cos2 x + b2 sin2 x d x,
(9.2.6)
π (w f + 1) 0
√
2 wf
2 wf + 1
a=
,
b=
.
wf + 1
wf + 1
9.2 Solutions for W F (w f ), W F (α f , w f ), and W S (ws )
301
Iwata chose to work with (9.2.6), using the representation of K as a hypergeometric function; thus
π
K ( a 2 cos2 x + b2 sin2 x) d x
0
=
∞ (2n)! 2 π 2
π
(a cos2 x + b2 sin2 x)n d x.
2
22n (n!)2
0
(9.2.7)
n=0
The integral on the right-hand side of (9.2.7) can be evaluated as an infinite sum,
and the double sum so produced was identified as F4 , an Appell hypergeometric
function of two variables; see Whittaker and Watson [60], p. 300. In this case a
theorem derived by Bailey [3] showed that this F4 is in fact the product of two
complete elliptic integrals. It is more straightforward to express the result in terms
of (9.2.5):
π
√
K ( A + B cos x) d x = 2K (k+ )K (k− ),
0
2
2k±
=1±
A2 − B 2 −
(1 − A)2 − B 2 .
(9.2.8)
This result may now be accepted as a standard form. Hence, for W F (w f ) we have
W F (w f ) =
2k 2f ±
=1±
4
π 2 (w
√
4 wf
f
K (k f + )K (k f − ),
(w f − 1)( w f − 3)
−
.
(w f + 1)3/2
+ 1)
(w f + 1)3/2
(9.2.9)
A further advance was made by Joyce [31], who published a generalization of
Iwata’s work. Thus he solved
π π π
d x d y dz
1
,
W F (α f , w f ) = 3
π 0 0 0 w f − α f cos x cos y − cos y cos z − cos z cos x
(9.2.10)
the solution being
W F (α f , w f ) =
where
4
K (k+ )K (k− ),
π 2 (w f + α f )
1/2
4α f w f
1
1
+
wfαf
(w f + α f )2
wf − αf
1/2
w f + (2 − α f )
w f − (2 + α f
−
2
(w f + α f )
2
2k±
(α f , w f ) = 1 ±
1/2
.
(9.2.11)
His analysis involved showing that W F (α f , w f ) could be expressed as
∞
J0 (at)J0 (bt)J0 (t) dt,
(9.2.12)
0
302
Seventy years of the Watson integrals
where a and b are functions of w f and α f and J0 is a Bessel function. A theorem
of Bailey [4] then enabled (9.2.12) to be evaluated directly. Joyce indicated that
he would describe the process in detail later but he never did. Instead, 32 years
later, under the prompting of colleagues the result was republished but was shown
to be derivable by Iwata’s method. We are assured that the promised other proof
will be forthcoming.
Rather interestingly, in his evaluation of W S , Watson [59] had employed the
same strategy as Iwata in evaluating his last integral using the method of Appell
and Bailey, but his result was confined to a special case. Also, Iwata ends his paper
with the remark that W S might have a similar result to (9.2.6). Now one can be
assured that Iwata must have tried his technique on W S (ws ), since the latter has a
considerably simpler form than W F (w f ), but if indeed one performs the first two
integrations one obtains
π π π
d x d y dz
1
W S (ws ) = 3
π 0 0 0 ws − cos x − cos y − cos z
π
2
2
1
K
d x.
(9.2.13)
= 2
ws − cos x
π 0 ws − cos x
This form was actually obtained by Tikson [56]. However, this does not yield to
Iwata’s approach and it seems that a different technique is required to obtain a
closed form for W S (ws ). This was accomplished by Joyce [32] in a classic paper,
and a brief description of his method is now given.
The procedure used by Joyce was entirely different to that of Iwata. He began
by considering the P form of the simple cubic,
π π π
3
d x d y dz
1
3
WS
.
=
PS (u) = 3
u
u
π 0 0 0 1 − 13 u (cos x + cos y + cos z)
(9.2.14)
By expanding the integrand in (9.2.14) in powers of u, a power series for P(u)
was found. Thus
∞
pn u n , |u| < 1,
(9.2.15)
PS (u) =
n=0
where
pn =
1
π3
0
π
0
π
π
0
1
3 (cos x
+ cos y + cos z)
2n
d x d y dz.
(9.2.16)
In the theory of random walks, the coefficient pn gives the probability that a random walker will return to his starting point (not necessarily for the first time) after
a walk of 2n steps on a SC lattice. For n → ∞ the behaviour of pn is described
by an asymptotic formula given by Domb [18], which showed that the range of
9.3 The Watson integrals between 1970 and 2000
303
validity of (9.2.15) could be extended to |u| = 1. So, PS (1), which is equal to
3W S , may be written as follows:
PS (1) =
∞
pn .
(9.2.17)
n=0
Joyce then proceeded to find a closed form for pn and to establish a three-term
recurrence relation amongst the pn . This enabled him to construct the following
third-order differential equation for PS :
d 3 PS
d 2 PS
2
+
12x(2x
−
15x
+
9)
dx3
dx2
d PS
+ 3(9x 2 − 44x + 12)
+ 3(x − 2)PS = 0,
dx
4x 2 (x − 1)(x − 9)
(9.2.18)
where x = u 2 . A remarkable result of Appell [1] allowed this third-order equation
to be reduced to a second-order equation, and the solutions of this were identified
as Heun functions. Then applying various standard transformation to these, they
were turned into hypergeometric functions and finally produced the important
result
√
4 − 3t
2
K (k+ )K (k− ),
W S (ws ) = 2
π ws 1 − t
√
√
2
2k±
= 1 ± 12 v 4 − v − 12 (2 − v) 1 − v,
(9.2.19)
'
2ws2 t = ws2 + 3 − (ws2 − 9)(ws2 − 1), v = t/(t − 1).
We have set out here in a few words an investigation which occupies almost
40 printed pages, and once again we suggest that the original paper be viewed in
order to appreciate the effort put into obtaining (9.2.19).
9.3 The Watson integrals between 1970 and 2000
In the years that followed Iwata’s solution for W F (w f ), several authors, e.g.,
Glasser [21], Hioe [28], Rashid [50], and Montaldi [46], investigated increasingly
complex forms of (9.1.11). All yielded to Iwata’s method. As an example we give
Glasser’s [21] form of (9.1.11):
φ = cos x cos y cos z+cos x cos y+cos y cos z+cos z cos x +cos x +cos y+cos z,
(9.3.1)
with solution
π π π
4
d x d y dz
1
= 2
K 2 (k),
3
π 0 0 0 w−φ
π (w + 1)
1
w − 7 1/2
where k 2 =
.
(9.3.2)
1−
2
w+1
304
Seventy years of the Watson integrals
What real lattice this represents is open to question, but it is a tribute to Iwata’s
approach that such a complex integral was evaluated by his method.
At around this time, Zucker stumbled accidentally into WIs and this led to his
first contribution. To explain how this occurred, more information on complete
elliptic integrals needs to be given. It had been shown by Abel that when
√
K (k)
a+b N
(9.3.3)
=
√ ,
K (k)
c+d N
with a, b, c, d, N all integers, then k is a root of an algebraic equation with integral coefficients
which can be solved in radicals. In the particular case where
√
K /K = N the k(N ) are called singular values, and for convenience K [k(N )]
will be denoted by K [N ]. It has long been known for N = 1, 3, and 4 that
K [N ] can be expressed in terms of gamma functions of rational arguments, the
values for K [1] and K [3] having been found by Legendre (see [60],
√ p. 524).
Now, the result given in (9.1.5) for W B contains the square of K (1/ 2), which
is indeed equal to K [1], and the result (9.1.6) for W F contains the square of
√
√
K ( 3 − 1)/2 2 which is K [3]. Of course Watson knew this; hence his translation of the results in terms of K into gamma functions. However, the K involved
in W S is actually K [6] and its expression in gamma functions
√ was unknown.
Glasser and Wood [23] had actually given K [2], for which k = 2 − 1, in terms
of gamma functions. This might have been extracted from a result of Ramanujan
[50],√
who had actually found the complete elliptic integral of the second kind for
k = 2 − 1 but no one seemed to have noticed. Now, in investigating properties
of the double sums
∞
(am 2 + bmn + cn 2 )−s ,
(m,n=0,0)
a comparatively straightforward procedure gave K [N ] in terms of gamma functions for many N ; see Zucker [61]. Indeed, K [N ] for N = 1, . . . , 16 (excluding
N = 14, which did not succumb to this technique) were thus evaluated. A paper
by Selberg and Chowla [54] proved that all K [N ] could – with sufficient labour –
be evaluated in terms of gamma functions, but they only gave explicit values for
N = 5 and N = 7. Thus the values found were new and included N = 6.
Together, Glasser and Zucker found [24]
√
4 6 1 5 % 7 & 11 (9.3.4)
W S = 2 24 24 24 24 ,
π
which unfortunately was wrong – a factor of 384π had been omitted. While it is
almost certain that everyone at some time has produced a wrong sign in a calculation or has lost a factor 2 somewhere, 384π requires an explanation. The reason
is depressingly simple. The result found for the square of K [6] was
9.3 The Watson integrals between 1970 and 2000
K 2 [6] =
305
% & √
√
√
√
1
5
7
24
24
( 2 − 1)( 3 + 2)(2 + 3) 24
11
24
384π
(9.3.5)
√
√
and,
was substituted
+ 12 2 − 10 3 −
√ this √
√
√into (9.1.9), the fact that (18√
√ when
7 6)( 2−1)( 3+ 2)(2+ 3) dramatically collapsed to 6 made the authors
forget the 384π in the denominator of (9.3.5) in the desire to publish quickly. So,
the correct result is
√
6 1 5 % 7 & 11 (9.3.6)
24 24 24 24 ,
WS =
96π 3
as is pointed out by everyone who quotes the original. Later, Borwein and Zucker
[6], making use of the surprising relation
1
11
24
√
√
24
% & = 3(2 + 3)1/2 ,
5
7
24
24
allowed the reduction of (9.3.4) to the even more compact form
√
( 3 − 1)
2
1
WS =
[( 24
)( 11
24 )] .
96π 3
(9.3.7)
A more substantial improvement to the theory was made by Joyce [33, 34] –
which considerably simplified Iwata’s result (9.2.9) for W F (w f ) and his own
expression (9.2.19) for W S (ws ). In both cases the original results appeared as the
products of two complete elliptic integrals with different moduli. In the abovereferenced papers it was shown that the results could be expressed as the square
of a single K . In [34], by using the elliptic modular transformation of order 3,
the solutions were found in a particularly simple fashion. They are given below in
parametric form:
2
4ξ(1 − 3ξ )(1 + ξ ) 2
K (k) ,
(9.3.8)
W F (w f ) =
(1 − ξ )3 (1 + 3ξ ) π
where
k2 =
and
16ξ 3
,
(1 − ξ )3 (1 + 3ξ )
ξ=
wf + 1 −
wf − 3 +
√
√
wf
wf
,
2
2
(1 − 9ξ 4 )
K (k)
W S (ws ) =
ws (1 − ξ )3 (1 + 3ξ ) π
with
16ξ 3
,
k2 =
(1 − ξ )3 (1 + 3ξ )
8
9
9 ws − w 2 − 1
:
s
ξ=
.
ws + ws2 − 9
(9.3.9)
306
Seventy years of the Watson integrals
Now define the sets of points in the w f and ws cut planes by C f and Cs respectively. There is insufficient space to expand upon the regions of C f and Cs over
which Iwata’s and Joyce’s original results are valid. Suffice it to say that their
range was severely limited in those sectors. The new results were valid for all C f
and Cs .
The complicated structures of (9.3.8) and (9.3.9) may be simplified by applying
various 2 F1 transformation formulae to the complete elliptic integrals in these
formulae. For example, Delves and Joyce [15] showed that
2
1 3
,
2ws − ws2 − 9
8 8 ; t (w )
,
(9.3.10)
W S (ws ) =
2 F1
s
1
ws2 + 3
where
t (ws ) =
2
'
16
2
2
2−9
w
(w
−
5)
−
(w
−
1)
w
.
s
s
s
s
(ws2 + 3)4
This result may also be used to calculate W S (ws ) at any point in Cs . For ws = 3
the original Watson integral becomes
2
1 3
,8 1
1
8
;
.
(9.3.11)
W S = 2 2 F1
1 9
Then making use of a famous result of Clausen [10], this may be written as
1 1 3
, 2, 4 1
1
4
;
W S = 2 3 F2
.
(9.3.12)
1, 1 9
From this and (9.3.7), we find the striking result that
√
∞
(4n)!
1
3 − 1 1 11 2
24 24
=
.
(n!)4 (48)2n−1
π3
(9.3.13)
n=0
9.4 The singly anisotropic simple cubic lattice
The equation
W S (αs , ws ) =
1
π3
0
π
0
π
π
0
d x d y dz
,
ws − αs cos x − cos y − cos z
(9.4.1)
is a further generalization of the original Watson integral W S . In terms of random
walks on a cubic lattice, the spacing between lattice points in one of the dimensions is different from that of the other two. It might be better to refer to (9.4.1)
as the Watson integral for a tetragonal lattice. The investigation of Delves and
Joyce [15] in finding a closed form for (9.4.1) is an exceptional work of analysis, which again can be described only briefly here. The integrand in (9.4.1) was
expanded in powers of 1/ws and the resulting series integrated term by term. Thus
9.4 The singly anisotropic simple cubic lattice
ws W S (αs , ws ) = y =
∞
μ2n (αs )z n ,
307
(9.4.2)
n=0
where z = 1/ws2 , and
π π π
1
μ2n = 3
(αs cos x + cos y + cos z)2n d x d y dz.
π 0 0 0
(9.4.3)
An explicit expression was found for μ2n (αs ) and a complex recurrence
relation established amongst μ2n+2 (αs ), μ2n (αs ), μ2n−2 (αs ), μ2n−4 (αs ), and
μ2n−6 (αs ). From this it was shown that y is the solution of a sixth-order differential equation L 6 (y) = 0, where L 6 is an extremely long and intricate operator.
These authors then showed it is possible to express L 6 as the product of a fourthorder differential operator, L 4 , and a second order differential operator and thus to
reveal that y is the solution of L 4 only. A further step then allowed the solutions
of L 4 (y) = 0 to be expressed as a product of the solutions of two second-order
differential equations. Then Schwarzian transformation theory enabled both these
second-order differential equations to be reduced to the standard Gauss hypergeometric differential equation. Finally, the hypergeometric functions which are
solutions of these equations could be transformed by well-known quadratic transformations into complete elliptic integrals as in (9.1.9). The final solution for y is:
ws W S (αs , ws ) = 2[1 − (2 − αs )2 z + 1 − (2 + αs )2 z]−1/2
2
2
K (k+ )
K (k− ) ,
×
(9.4.4)
π
π
where
2
2
2k±
≡ 2k±
(αs , z)
−3
1 − (2 − αs )2 z + 1 − (2 + αs )2 z
=1−
'
'
√
√
×
1 + (2 − αs ) z 1 − (2 + αs ) z
'
'
√
√
+ 1 − (2 − αs ) z 1 + (2 + αs ) z
'
'
'
√
√
× ±16z + 1 − αs2 z 1 + (2 − αs ) z 1 − (2 + αs ) z
+
'
'
√
1 − (2 − αs ) z 1 + (2 + αs ) z .
√
(9.4.5)
This is a few-word summary of some 60 pages of analysis and, for the third
time here, we recommend readers to view the original paper to appreciate the
tenacity that went into producing (9.4.4) and (9.4.5), to which the comment of
Bornemann et al. [5] may be added: ‘What a triumph of dedicated men; for such
problems current computer algebra systems are of little help’.
308
Seventy years of the Watson integrals
Is there a more direct approach to (9.4.5)? It turns out that there is, but it is
doubtful whether it would have been found without the inspiration of the known
result. Going back to Tikson’s result (9.2.12) and putting in the anisotropy does
not complicate the last integral very much. In fact, doing the first two integrals
using Iwata’s approach, one obtains
π
2
2
1
K
d x.
(9.4.6)
W S (αs , ws ) = 2
ws − αs cos x
π 0 ws − αs cos x
Now the following substitution is made,
cos x =
ws cos ψ + αs
,
ws + αs cos ψ
and a much simplified version of (9.4.6) appears, namely
π
2
1
W S (αs , ws ) = K (C + D cos ψ) dψ,
ws2 − αs2 π 2 0
(9.4.7)
(9.4.8)
where
C=
2ws
− αs2
ws2
and
D=
2αs
.
ws2 − αs2
(9.4.9)
In going through the first two steps of the Iwata process for the anisotropic facecentred cubic lattice, we arrived at
π
√
2
1
W F (α f , w f ) =
K ( A + B cos x) d x,
(9.4.10)
2
wf + αf π 0
where
A=
2(2α f w f + 1)
(w f + α f )2
and
B=
2
,
(w f + α f )2
(9.4.11)
and the similarity between (9.4.8) and (9.4.10) is evident. We have of course the
solution of (9.4.10) in (9.2.11), and it seemed natural to look for a solution to
(9.4.8) along the same lines as Iwata had done for (9.4.10). This was accomplished
as follows. For convenience we write
π
√
2
K ( A + B cos x) d x,
I (A, B) = 2
π 0
π
2
J (C, D) = 2
K (C + D cos ψ) dψ.
(9.4.12)
π 0
As Iwata did, the integrand of I (A, B) is expanded and integrated term by term
to yield a double series. However, instead of identifying this double series as an
Appell hypergeometric function, I (A, B) is represented by a Kampé de Feriet
series. Doing the same thing to J (C, D) yields a different Kampé de Feriet series
but, by applying appropriate transformation formulae to the I (A, B) series, it can
9.4 The singly anisotropic simple cubic lattice
309
be put into the form for J (C, D). The following remarkable connection formula
was found:
J (C, D) = (1 − A + B)1/2 I (A, B),
(9.4.13)
where A and B are appropriate solutions of the simultaneous equations
C2 = −
(A − B)(1 − A + B)
,
(1 − A + B)2
D2 =
2B
.
(1 − A + B)2
(9.4.14)
By finding relevant solutions for A and B and substituting into (9.4.13), after
considerable algebraic simplification a standard form for J (C, D) may now be
established. It is
2
2
−1/2 2
2
2
K (k+ )K (k− ),
J (C, D) = 2[(1 − D) − C + (1 + D) − C ]
π
(9.4.15)
where
−3
1 (1 − D)2 − C 2 + (1 + D)2 − C 2
2k± = 1−
C × 2 (C + D) 1 − (C − D)2 + (C − D) 1 − (C + D)2
,
× ±4C C 2 − D 2 +
(1 + C)2 − D 2 + (1 − C)2 − D 2 .
(9.4.16)
This can now take its place as a standard integration alongside the result for
I (A, B). If the values of C and D given in (9.4.9) are substituted in the above, the
result obtained for W S (αs , ws ) is precisely the same as that given by Delves and
Joyce [15].
A further consequence of the relationship between I (A, B) and J (C, D) is that
a connection between W F (α f , w f ) and W S (αs , ws ) may be found. Thus
2αs
W F (α f , w f ) = − W S (αs , ws ), (9.4.17)
2
2
ws − (2 + αs ) + ws2 − (2 − αs )2
with
wf = −
'
'
1
(ws2 − 4 − αs2 ) + ws2 − αs2
ws2 − (2 + αs )2
4αs
'
'
'
+ ws2 − (2 − αs )2 + ws2 − (2 + αs )2 ws2 − (2 − αs )2
(9.4.18)
and
'
'
1
2
2
2
2
(ws − 4 − αs ) − ws − αs
αf = −
ws2 − (2 + αs )2
4αs
'
'
'
2
2
2
2
2
2
+ ws − (2 − αs ) + ws − (2 + αs ) ws − (2 − αs ) .
(9.4.19)
310
Seventy years of the Watson integrals
The following inverse connection formula can also be found:
α f (w f + 2 − α f )(w f − 2 − α f ) 1/2
ws W S (αs , ws ) =
w f W F (α f , w f ),
w f (α f w f + 1)
(9.4.20)
where
α f w f (w f + 2 − α f )(w f − 2 − α f )
wf + αf
. (9.4.21)
,
αs =
ws2 = −
2
αfwf + 1
(α f w f + 1)
Thus we require either the result for W F (α f , w f ) or the result for W S (αs , ws )
for the other to be determined. All is explained in detail in a paper of Joyce et al.
[37]. The ‘final problem’ yet to be solved is
π π π
d x d y dz
1
. (9.4.22)
W S (αs , βs , ws ) = 3
π 0 0 0 ws − αs cos x − βs cos y − cos z
The latter might be referred to as the Watson integral for the doubly anisotropic
cubic lattice or the Watson integral for an orthorhombic lattice. The difficulties encountered so far in attempted solutions of this three-parameter problem
seem insuperable. Even attempts at reducing it to a Montroll-type two-parameter
exercise, namely
π π π
d x d y dz
1
,
W S (1 + αs + βs ) = 3
π 0 0 0 1 + αs + βs − αs cos x − βs cos y − cos z
(9.4.23)
have not yielded any success.
9.5 The Green’s function of the simple cubic lattice
Recently some significant progress has been made in finding exact product forms
for certain forms of
π π π
cos lx cos my cos nz
1
d x d y dz.
G(l, m, n; αs , ws ) = 3
π 0 0 0 ws − αs cos x − cos y − cos z
(9.5.1)
A series of papers by Joyce and Delves [35, 36] and Delves and Joyce [16, 17]
includes extensive detail in the presentation of results, which are briefly summarized here.
These new results depend on a beautiful extrapolation of Iwata’s result for
I (A, B). To see how this extrapolation is best presented, consider
π
1 1
√
1 π
2
2 , 2 ; A + B cos x d x.
K ( A + B cos x) d x =
I (A, B) = 2
2 F1
π 0
1
π 0
(9.5.2)
This may be taken to be a special case of
t, 1 − t
1 π
In (t; A, B) =
; A + B cos x cos nx d x,
(9.5.3)
2 F1
1
π 0
9.5 The Green’s function of the simple cubic lattice
311
where clearly Iwata’s I (A, B) is I0 ( 12 ; A, B). Now, Delves and Joyce [17] found
a closed form for In (t; A, B), namely
2 √
(t)n (1 − t)n 1 √
In (t; A, B) =
1− A− B− 1− A+ B
2B
(n!)2
t, 1 − t
t, 1 − t
; θ+ 2 F1
; θ− ,
(9.5.4)
× 2 F1
n+1
n+1
where (t)n is the Pochhammer symbol, given by (t + n)/ (n), and
2θ± = 1 ± A2 − B 2 − (1 − A)2 − B 2 .
(9.5.5)
The remarkable expression (9.5.4) was first obtained using a Lie-group addition
formula applicable to 2 F1 first obtained by Miller [45]. It was later confirmed by
generalizing Iwata’s original approach. For some particular choices of (l, m, n)
the Fourier transform of the Green’s function may be reduced to an elliptic integral. Some of these elliptic integrals correspond to Miller’s form, and the Green’s
function can be evaluated in a few pages of hand calculation. For example,
W S (2n, n, n; αs , ws ) was shown to be given by
1/2
1
W S (2n, n, n; αs , ws ) =
ws2 + 4 − αs2
1 3
2
2
2
1 π
4 , 4 ; 8 2ws − αs + αs cos x cos nx d x.
×
2 F1
π 0
1
(ws2 + 4 − αs2 )2
(9.5.6)
(It is noteworthy that in obtaining (9.5.6) the authors met a complex trigonometric integral whose solution as an elliptic integral was found in Jacobi [30].
An alternative procedure originating in Cayley [8, 9] was combined with an 2 F1
transformation of Goursat [25]. When stuck go to the experts.) Equation (9.5.6) is
clearly of the form (9.5.4) and thus the following closed form is obtained:
3
1/2 1
4 n 4 n
1
W S (2n, n, n; αs , ws ) =
ws2 + 4 − αs2
(n!)2
'
2 2n
'
1
2
2
2
2
×
ws − (2 − αs ) − ws − (2 + αs )
8αs
× 2 F1
1
4 n
n+1
; η+
2 F1
1 3
4, 4
n+1
; η− ,
(9.5.7)
3
4 n
are Pochhammer symbols and
'
ws
±16
2η± = 1 + 2
ws2 − αs2
(ws + 4 − αs2 )2
'
'
− (ws2 − 4 − αs2 ) ws2 − (2 − α)2 ws2 − (2 + α)2 .
where
and
1 3
4, 4
(9.5.8)
312
Seventy years of the Watson integrals
Similarly,
ws W S (n, n, n; 1, ws ) =
1
π
gives the closed form
ws W S (n, n, n; 1, ws ) =
π
2 F1
1
0
1
3 n
2
3 n
(n!)2
× 2 F1
with
1 2
3, 3
;
27
(ws + cos x) cos nx d x
4ws3
(9.5.9)
3n
'
1
2−9
−
w
w
s
s
3
1 2
3, 3
n+1
; ξ+
2 F1
1 2
3, 3
n+1
; ξ− ,
)
)
1
9
1
ξ± (ws ) =
4ws2 + (9 − 4ws2 ) 1 − 2 ± 27 1 − 2
2
8ws
ws
ws
.
(9.5.10)
(9.5.11)
Interestingly, if in (9.5.7) and (9.5.10) n is made zero and ws is set equal to 3, two
new formulae for the original Watson integral W S may be obtained. Along with
(9.1.9) we thus have
2
1 1
√
√ 2
√ √
√
√
,
W S = (18 + 12 2 − 10 3 − 7 6) 2 F1 2 2 ; {(2 − 3)( 3 − 2)}
1
2
2
1 3
1 3
√
,
,
1
1
8 8;1
4 4 ; 1 ( 2 − 1)2
=
=
2 F1
2 F1
2
1 9
2
1 6
√
√
2
1 2
√ 2
2
,3 1
3 − 1 1 11 2
3
; 4 (2 − 2)
24 24
=
=
.
2 F1
3
1
96π 3
(9.5.12)
We believe it was Littlewood who once remarked ‘All equations are identities.’
Does (9.5.12) support or give the lie to this statement?
9.6 Generalizations and recent manifestations
of Watson integrals
An obvious generalization of WIs is to higher dimensions. A recent survey by
Guttmann [26] is an excellent source of information on these objects. Guttmann
notes that for two-dimensional lattices WIs are given in terms of a single K .
For three-dimensional lattices the solutions appear as products of two complete
elliptic integrals K (k+ )K (k− ). It appears that this occurs because the underlying
third-order ordinary differential equation (ODE) obeyed by these lattices has the
almost-magical Appell reduction property, allowing their solution to be expressed
in terms of an associated second-order ODE. For four-dimensional lattices it
appears that the underlying ODEs are all of the Calabi–Yau type. This is true for
all five-dimensional lattices as well, except for the five-dimensional FCC lattice.
9.6 Generalizations and recent manifestations of Watson integrals
313
Let us consider the four-dimensional simple cubic lattice in more detail. We
have
π π π π
d x d y dz dt
1
W S,4d (ws ) = 4
. (9.6.1)
π 0 0 0 0 ws − cos x − cos y − cos z − cos t
Its P-form is
π π π π
d x d y dz dt
1
4
π 0 0 0 0 1 − (u/4)u (cos x + cos y + cos z + cos t)
4
4
= WS
.
(9.6.2)
u
u
PS,4d (u) =
It has been shown that the d-dimensional diamond (D) lattice is simply related to
the (d + 1)-dimensional hypercubic lattice via an Abel transform – see Guttmann
and Prellberg [27], and Glasser and Montaldi [22]. The D lattice has not been
previously discussed, since in three dimensions the analysis involved is identical
to that in the FCC case. In terms of their respective P functions we have
4
3
1
1
PD,3d
=
PF,3d
.
(9.6.3)
4u + 4
1+u
4u
u
As can be seen, when u = 3 the two are identical apart from a numerical factor.
Since everything about the three-dimensional diamond lattice is known, this suggests an interesting approach to the four-dimensional simple cubic lattice. Indeed
it may be shown that the four-dimensional simple cubic P form may be expressed
as a single integral (Guttmann [26]) namely
8 1 K (k+ )K (k− )
dt
PS,4d (u) = 3
√
π 0
1 − t2
where
k± =
1
2
± 14 u 2 t 2 4 − u 2 t 2 − 14 (2 − u 2 t 2 ) 1 − u 2 t 2 .
(9.6.4)
A single integral for W S,4d (ws ) was obtained by Zucker by doing two integrals
following Iwata and a third integral following Joyce and Zucker [39], leading to
π 2 2 K (k ) p (1 − p 2 )(1 − 9 p 2 )
8
dt,
(9.6.5)
W S,4d (ws ) = 3
π 0
(1 − p)3 (1 + 3 p)
where
γ = ws − cos t,
p =
2
γ−
γ+
γ2 − 1
γ2 − 9
,
and
k2 =
16 p 3
.
(1 − p)3 (1 + p)
Both (9.6.4) and (9.6.5) easily provide numerical values for any ws and u, but one
is no nearer to a closer-form evaluation.
314
Seventy years of the Watson integrals
It has already been noted, in Section 9.2 that Bessel functions are connected
to WIs. For example, it is simple to express W S as a single integral as follows.
First, put
∞ π π π
1
WS = 3
et (−3+cos x+cos y+cos z) d x d y dz dt;
(9.6.6)
π 0
0
0
0
since
1
π
π
et cos x d x = I0 (t),
(9.6.7)
0
we then have
WS =
0
∞
e−3t I03 (t) dt,
(9.6.8)
where I0 (t) is a modified Bessel function of the first kind. Now, in a recent
investigation, Bailey et al. [2] studied integrals of the form
∞
∞
k n
t K 0 (t) dt,
and
tn,2k+1 =
t 2k+1 I02 (t)K 0n−2 (t) dt
cn,k =
0
0
(9.6.9)
amongst many others. Here K 0 is a modified Bessel function of the second kind
and should not be confused with a complete elliptic integral of the first kind.
These integrals arose from certain Feynman diagrams and connections with WIs
in three dimensions have been found, though some are still conjectures. It has
been established that
∞
π3
K 03 (t) dt =
(9.6.10)
W F (3)
c3,0 =
2
0
and conjectured that
∞
t5,1 =
0
?
t I02 (t)K 03 (t) dt = π 2 W F (15).
Also, Broadhurst [7] has proved that
∞
π2
W F (3).
t Io2 (t)K 02 (t)K 0 (2t) dt =
12
0
(9.6.11)
(9.6.12)
This may just be the tip of an iceberg and possibly many other similar results
remain to be discovered.
Another appearance of WIs is in the field of Mahler measures. The connection
arises since the integrals may be expressed as a derivative with respect to the
appropriate complex variable w of a logarithmic form. For example,
π π π
1
d
log(w
−
cos
x
−
cos
y
−
cos
z)
d
x
d
y
dz
.
W S (ws ) =
s
dws π 3 0 0 0
(9.6.13)
9.7 Commentary: Watson integrals and localized vibrations
315
These logarithmic integrals are involved in the calculation of the total number of
spanning trees on a hypercubic lattice (Rosengren [53]) and in the theory of collapsing branched polymers (Madras et al. [41]). A rapid way of evaluating these
integrals in all dimensions was devised by Joyce and Zucker [38]. The definition
of a Mahler measure m of an n-variable polynomial P(z 1 , . . . z n ) is (here P is not
to be confused with a modified WI)
∞
∞
m [P(z 1 , . . . , z n )] =
...
log |P e2πiθ1 , . . . e2πiθn | dθ1 · · · dθn .
0
0
(9.6.14)
It was noted by Rogers [52] that two Mahler measures in which he was interested,
namely
g1 (u) = m u + x + x −1 + y + y −1 + z + z −1
and
g2 (u) = m − u + 4 + (x + x −1 )(y + y −1 )
+ (y + y −1 )(z + z −1 ) + (z + z −1 )(x + x −1 )
(9.6.15)
were closely related to the SC and FCC versions of WIs. This came as a surprise
to Rogers as much as it did to some authors who were working on WIs. The
appearance of WIs in these unexpected places shows their ubiquitous nature, and
this seems an appropriate juncture to end this review.
9.7 Commentary: Watson integrals and localized vibrations
(1) One way of thinking about WIs is that they provide a connection between
lattice sums, periodic Green’s functions, and single-source Green’s functions.
Lattice sums themselves can be regarded as Green’s functions. If we go back
to Chapter 3, the Green’s function of Section 3.3 was defined as a lattice sum
for the generalized Poisson equation, with a set of sources at the points S = R p :
⎡
⎤
s/2 = i s
eik·S = i s (ab) ⎣
δ(S − R p ) − 1⎦ .
(9.7.1)
k=0
Rp
We can convert
from a multiple-source Green’s function to a single-source
Green’s function by multiplying it by a suitable weight function and integrating
over the unit cell of S. We use the property that
(S)e−ik·Rq d 2 S = 0
unless S = Rq .
(9.7.2)
UC
The single source chosen is then at S = Rq . A set of sources can be picked out
using other weight functions. The Watson integral (9.5.1) is a localized Green’s
function of this type.
316
Seventy years of the Watson integrals
(2) A recent article by Colquitt et al. [11] treats a two-dimensional vibrational
problem in this fashion. In the case of N defects in an infinite square lattice, the
Fourier-transformed equation for vibrations of angular frequency ω in the lattice
is written as
(ω2 − 4 + 2 cos ξ1 + 2 cos ξ2 )u(ξ ) =
N
−1
u p,0 e−i pξ1 .
(9.7.3)
p=0
This equation has oscillating solutions in ω2 < 8, but for ω2 > 8 the solution
decays exponentially with increasing distance from the defect region. The interest
here is in the behaviour of the solution in the exponentially decaying or stop-band
frequency region.
The field is written in terms of linear combinations of N shifted Green’s
matrices, defined as
π π
cos (n 1 − p)ξ1 cos n 2 ξ2
1
dξ1 dξ2 .
(9.7.4)
g(n, p; ω) = 2
2
π 0 0 ω − 4 + 2 cos ξ1 + 2 cos ξ2
These of course are the doubly periodic form of (9.5.1). Letting a = ω2 /2 − 2 +
cos ξ2 , the Green’s matrix may be reduced to a single integral:
π √ 2
1
( a − 1 − a)|n 1 − p|
g(n, p; ω) =
dξ2 .
(9.7.5)
√
2π 0
a2 − 1
An alternative representation is in a Bessel-function product form, with two elements rather than the three occurring in (9.2.12). Also, instead of the oscillating
function J0 we have the modified function I0 , given that we are assuming ω lies
in the band gap:
(−1)n 1 − p+n 2 ∞
In 1 − p (x)In 2 (x) d x.
(9.7.6)
g(n, p; ω) =
2
0
One way of obtaining a shifted WI is to start with the elliptic integral form for the
unshifted WI,
1
4
g(0, 0, 0; ω) =
,
(9.7.7)
K
aπ
a2
and use repeated integration by parts together with the recurrence relation In (x) =
(x) − I
2In−1
n−2 (x). Colquitt et al. [11] also give a representation for g(n, p; ω)
in terms of the hypergeometric function 4 F3 .
9.8 Commentary: Variations on W S
As recounted already in this chapter, much effort has been put into evaluating the
integral
π π π
d x d y dz
1
(9.8.1)
W = 3
π 0 0 0 3 − cos x − cos y − cos z
9.8 Commentary: Variations on W S
and its generalization
W (w) := W S (w) =
1
π3
π
0
0
π
π
0
317
d x d y dz
.
w − cos x − cos y − cos z
(9.8.2)
For amusement (and reference) we list here solutions of W (w), in terms of singular values of complete elliptic integrals of the first kind, K (k 2 ), which have been
discovered by various investigators. As in Section 1.12 we use
√
b( p) = β( p) tan pπ
where
β( p) =
2 ( p)
(2 p)
and give W (w) for various values of w along the real axis.
2 1
b
√
3
W (0) = −21/3 3
i,
2
12π
2 1
√ b 8
,
W (1) = (1 − 2i)
16π 2
1
b2 20
√
,
W ( 5) = 21/5 (51/4 − 5−1/4 i)
80π 2
1
b2 24
2/3
,
W (3) = 2
96π 2
b2 13
√
3√
W
6 = 25/6 3
,
2
48π 2
b2 18
W (5) =
,
48π 2 1
11
17
b 120
b 120
b 120
W (9) = 213/15
,
% 7 &
480π 2 b 120
√ ⎡ b 1 b 25 ⎤2
5/21
168
168
7
2
⎦ ,
W (15) = 5 2 2 ⎣
37
2 3 7
πb
168
√ ⎡ b 1 b 25 b 35 b 49 ⎤2
2/3
312
312
312
312
13
2
⎦ ,
W (51) = 5 2 2 ⎣
13
23
61
2 3 13
π b 312 b 312 b 312
√
22/3 3
W (99) =
816
1
25
35
41
49
59
65
83
91
b 408
b 408
b 408
b 408
b 408
b 408
b 408
b 408
b 408
% & ×
.
5
7
13
29
31
47
79
b 408
b 408
b 408
b 408
b 408
b 408
π 2 b 408
318
Seventy years of the Watson integrals
Another set of integrals which may also be solved in terms of singular values is
π π π
1
α d x d y dz
J (α) := 3
.
(9.8.3)
π 0 0 0 2α + 1 − α cos x − α cos y − cos z
The general solution of this may be shown to be
(1 − 2k − k 2 )2
.
8k(1 − k 2 )
√
For some values of α the value of k found is such that K (k 2 ) = N K (k 2 ), where
N is an integer. In this case the solution to (9.8.3) is
(1 − 2k − k 2 )K (k 2 )K (k 2 )
J (α) =
√
where
α :=
K (k 2 [N ])
N (1 − 2k[N ] − k [N ])
π
2
2
,
and we thus obtain solutions in terms of singular values (see appendix section A.3). Thus, for α = 1, 4, 24, 49, 4900 we have N = 6, 10, 18, 22, 58 and
hence
⎡ ⎤2
1
π π π
2/3
2 ⎣ b 24 ⎦
d x d y dz
1
=
,
J (1) = 3
96
π
π 0 0 0 3 − cos x − cos y − cos z
π π π
1
4 d x d y dz
J (4) = 3
π 0 0 0 9 − 4 cos x − 4 cos y − cos z
⎡ ⎤2
1
9
1 ⎣ b 40 b 40 ⎦
=
,
80
πb 3
8
J (24) =
J (49) =
=
J (4900) =
=
×
1
π3
1
π3
π
π
0
π
π
0
π
0
π
√ ⎡ b 1 ⎤2
24 d x d y dz
6⎣ 8 ⎦
=
,
49 − 24 cos x − 24 cos y − cos z
24
π
49 d x d y dz
0
0
0 99 − 49 cos x − 49 cos y − cos z
⎤2
⎡ 1
9
19
3/22
b
b
88
88 b 88
2
⎣
% & ⎦ ,
5
7
176
b 88
π b 88
π π π
1
4900 d x d y dz
,
3
π 0 0 0 9801 − 4900 cos x − 4900 cos y − cos z
228/29 35
232
⎡ ⎤2
1
9
25
33
35
49
51
57
b 232
b 232
b 232
b 232
b 232
b 232
b 232
b 232
⎣
⎦ .
% & 5
7
13
23
45
53
b 232
b 232
b 18 b 232
b 232
b 232
π b 232
9.9 Commentary: Computer algebra
319
The value J (4900) is surely one of the most remarkable closed forms obtained in
the subject.
9.9 Commentary: Computer algebra
The comment in [5] that for some problems ‘current computer algebra systems are
of little help’ is slightly outdated. Such is the progress made in the last 5–10 years
that a modern computer algebra system such as Maple can now do an enormous
amount of work previously done using pencil and paper.
For instance, if one knows the recurrence satisfied by a sequence (say (9.2.16))
then the differential equation satisfied by its generating function (in our example,
(9.2.18)) can be obtained effortlessly using the command rectodiffeq from
the Maple package gfun.
Moreover, for many sums depending on a parameter n, a recurrence in n
can be automatically found (and provably certified) using the Wilf–Zeilberger
algorithm; these sums include a large number of binomial sums and many
hypergeometric identities. Analogously, the Almqvist–Zeilberger algorithm can
produce differential equations in t for integrals that depend on a parameter t. A
very readable treatment is given in Petkovsek et al. [48], and implementations of
these algorithms can be downloaded for free.
We return to the differential equation (9.2.18). In Maple 13 (and newer versions), if one simply applies the command dsolve to it then very quickly a
solution in terms of Heun functions is returned. This is possible because this particular third-order differential equation is a symmetric square (that is, its solutions
are the products of the solutions of a second-order differential equation), a fact
which can be readily checked; Maple then tries a variety of tricks to solve the pertinent second-order differential equation, including the application of a number of
algebraic transformations.
For (9.4.3), we mention that the generating function is annihilated by a
sixth-order differential operator, which can be written as a product of a fourthorder and a second-order differential operator. Factorizations of this sort can be
painlessly accomplished using the DFactor command from the Maple package
DETools.
Moreover, it is no longer necessary to perform hypergeometric or gamma function calculations by hand (as in (9.5.6)), since computer algebra systems are
substantially faster and more reliable and they can be ‘taught’ key values and
transformations for these functions.
Finally, we remark that oversights such as being out by a factor of 384π in
(9.3.4) can be greatly reduced when one uses a computer algebra system. First,
one may save all the main results (and their numerical values) on the same
worksheet and therefore numerical discrepancies may be spotted at a glance. Transcription errors can also be reduced owing to commands which convert the content
320
Seventy years of the Watson integrals
of a worksheet into LATEX. Suppose, however, that a numerical discrepancy has
crept in and one suspects to be out by a factor F from the true answer. If F is
likely to be a product of well-known constants (such as fractions, surds, multiples
or powers of π ), then one could either use the Maple command identify on
F or run the PSLQ algorithm on the vector consisting of log F as well as the
logs of any suspected constants – such as log 2, log 3, log π – in an attempt to
recover F.
It is not knowledge, but the act of learning, not possession but the act of getting there,
which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I
turn away from it, in order to go into darkness again; the never-satisfied man is so strange
if he has completed a structure, then it is not in order to dwell in it peacefully, but in order
to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is
scarcely conquered, stretches out his arms for others.
Carl Friedrich Gauss
From an 1808 letter to Farkas Bolyai (father of Janos Bolyai)
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Appendix
A.1 Tables of modular equations
Table A.1 Modular equations of order 3 and dimensions 2 and 4, each forming a set of four elements.
Ramanujan
Classical
Lambert series
4qψ(q 2 )ψ(q 6 )
θ2 θ2 (q 3 )
4
φ(q)φ(q 3 ) − 1
θ3 θ3 (q 3 ) − 1
2
∞ χ
n
−6a q
2n
1−q
1
∞
χ −3 q n
1 + (−q)n
1
φ(−q)φ(−q 3 ) − 1
θ4 θ4 (q 3 ) − 1
−2
∞ χ
n
−6b q
1 + qn
Mellin transform
4(1 − 2−2s )L 1 L −3
2(1 + 21−2s )L 1 L −3
−2(1 − 22−2s )L 1 L −3
1
4qψ(−q 2 )ψ(−q 6 )
θ5 θ5 (q 3 )
16qψ 2 ψ 2 (q 3 )
θ22 (q 1/2 )θ22 (q 3/2 )
φ 2 (q)φ 2 (q 3 ) − 1
θ32 θ32 (q 3 ) − 1
φ 2 (−q)φ 2 (−q 3 ) − 1
θ42 θ42 (q 3 ) − 1
16qψ 2 (−q)ψ 2 (−q 3 )
θ52 (q 1/2 )θ52 (q 3/2 )
∞ χ
n
12 q
2n
1+q
0
∞ χ
n
3a nq
16
2n
1−q
1
∞
χ 3a 4nq 4n
χ 6a nq n
4
+
1 − qn
1 − q 4n
1
∞
χ 3a 4nq 4n
χ 6a nq n
−4
−
1 + qn
1 − q 4n
4
4L −4 L 12
16(1 − 2−s )(1 − 31−s )L 1 (s)L 1 (s − 1)
4(1 − 21−s + 22−2s )(1 − 31−s )L 1 (s)L 1 (s − 1)
−4(1 − 22−s )(1 − 31−s )L 1 (s)L 1 (s − 1)
1
16
∞ χ χ
n
3a 2b nq
2n
1−q
1
16(1 − 2−s )(1 − 22−s )(1 − 31−s )L 1 (s)L 1 (s − 1)
326
Appendix
Table A.2 Modular equations of order 3 and dimension 2 forming a set of eight
elements
Ramanujan
ψ 3 (q)
−1
ψ(q 3 )
φ 3 (q)
−1
φ(q 3 )
φ 3 (−q)
−1
φ(−q 3 )
ψ 3 (−q)
−1
ψ(−q 3 )
ψ 3 (q 3 )
−1
ψ(q)
Classical
3
1/2
1 θ2 q
%
& −1
4 θ2 q 3/2
θ33 (q)
θ3 (q 3 )
θ43 (q)
θ4 (q 3 )
θ33 (q 3 )
φ 3 (−q 3 )
−1
φ(−q)
θ43 (q 3 )
ψ(−q)
−1
θ3 (q)
3
∞ χ
n
−6a q
1 − qn
Mellin transform
3(1 + 2−s )L 1 L −3
1
∞ χ
n
−6b q
6
1 + (−q)n
6(1 + 21−s − 21−2s ))L 1 L −3
1
−1
3
1/2
1 θ5 q
%
& −1
4 θ5 q 3/2
3
3/2
1 θ2 q
%
& −1
4 θ2 q 1/2
φ 3 (q 3 )
−1
φ(q)
ψ 3 (−q 3 )
−1
Lambert series
−1
−6
∞ χ
n
−3 q
1
−3
∞ χ
n
−6a q
n
1+q
θ4 (q)
3
3/2
1 θ5 q
%
& −1
4 θ5 q 1/2
−6(1 − 21−s )L 1 L −3
−3(1 + 2−s )(1 − 21−s )L 1 L −3
1
∞ χ
n
−3 q
1 − q 2n
(1 − 2−s )L 1 L −3
1
−2
∞
1
−1
1 + qn
2
χ −3 q n
1 − (−q)n
∞ χ
n
−6b q
1 − qn
−2(1 − 21−s − 21−2s )L 1 L −3
2(1 + 21−s )L 1 L −3
1
∞ χ
n
−6b q
2n
1−q
1
(1 − 2−s )(1 + 21−s )L 1 L −3
Table A.3 Modular equations of order 3 and dimension 6 forming a set of eight elements;
A = L 1 (s)L −3 (s − 2), B = L 1 (s − 2)L −3 (s)
Ramanujan
Classical
Lambert series
64 qψ 5 (q)ψ(q 3 ) − 9q 2 ψ(q)ψ 5 (q 3 )
θ25 (q 1/2 )θ2 (q 3/2 ) − 9θ2 (q 1/2 )θ25 (q 3/2 )
64
∞ χ
2 n
−3 n q
1 − q 2n
Mellin transform
64(1 − 2−s )A
1
9φ(q)φ 5 (q 3 ) − φ 5 (q)φ(q 3 ) − 8
9θ3 θ35 (q 3 ) − θ35 θ3 (q 3 ) − 8
8
∞ χ
2 n
−3 n q
1 − (−q)n
1
9φ(−q)φ 5 (−q 3 ) − φ 5 (−q)φ(−q 3 ) − 8
64 qψ 5 (−q)ψ(−q 3 ) + 9q 2 ψ(−q)ψ 5 (−q 3 )
9θ4 θ45 (q 3 ) − θ45 θ4 (q 3 ) − 8
θ55 (q 1/2 )θ5 (q 3/2 ) + 9θ5 (q 1/2 )θ55 (q 3/2 )
64 qψ 5 (q)ψ(q 3 ) − q 2 ψ(q)ψ 5 (q 3 )
θ25 (q 1/2 )θ2 (q 3/2 ) − θ2 (q 1/2 )θ25 (q 3/2 )
φ 5 (q)φ(q 3 ) − φ(q)φ 5 (q 3 )
θ35 θ3 (q 3 ) − θ3 θ35 (q 3 )
φ(−q)φ 5 (−q 3 ) − φ 5 (−q)φ(−q 3 )
θ4 θ45 (q 3 ) − θ45 θ4 (q 3 )
−8
64
64
∞ χ
2 n
−6b n q
1 − qn
1
∞ χ
1
∞
1
64 qψ 5 (−q)ψ(−q 3 ) + q 2 ψ(−q)ψ 5 (−q 3 )
θ55 (q 1/2 )θ5 (q 3/2 ) + θ5 (q 1/2 )θ55 (q 3/2 )
2 n
−6b n q
1 − q 2n
n
2n
χ −6a q + q
(1 − q n )3
∞
2n
n
χ −3 q − (−q)
3
1 + (−q)n
1
∞
n
2n
χ −3 q − q
8
n
(1 + q )3
8
64
1
∞
1
n
2n
χ −6a q − q
n
(1 − q )3
8(1 + 21−s )(1 − 22−s ))A
−8(1 + 23−s )A
64(1 − 2−s )(1 + 23−s )A
64(1 + 2−s )B
8(1 − 21−s )(1 + 22−s )B
8(1 − 23−s )B
64(1 + 2−s )(1 − 23−s )B
Table A.4 Modular equations of order 5 and dimension 4 forming a set of eight elements;
C = L 1 (s)L 5 (s − 1), D = L 1 (s − 1)L 5 (s)
Ramanujan
Classical
16 qψ 3 (q)ψ(q 5 ) − 5q 2 ψ(q)ψ 3 (q 5 )
θ23 (q 1/2 )θ2 (q 5/2 ) − 5θ2 (q 1/2 )θ23 (q 5/2 )
5φ(q)φ 3 (q 5 ) − φ 3 (q)φ(q 5 ) − 4
5θ3 θ33 (q 5 ) − θ33 θ3 (q 5 ) − 4
5φ(−q)φ 3 (−q 5 ) − φ 3 (−q)φ(−q 5 ) − 4
5θ4 θ43 (q 5 ) − θ43 θ4 (q 5 ) − 4
Lambert series
∞ χ
n
5 nq
16
1 − q 2n
1
∞
χ 5 nq n
4
1 − (−q)n
1
16 qψ 3 (−q)ψ(−q 5 ) + 5q 2 ψ(−q)ψ 3 (−q 5 )
16 qψ 3 (q)ψ(q 5 ) − q 2 ψ(q)ψ 3 (q 5 )
φ 3 (q)φ(q 5 ) − φ(q)φ 3 (q 5 )
θ53 (q 1/2 )θ5 (q 5/2 ) + 5θ5 (q 1/2 )θ53 (q 5/2 )
θ23 (q 1/2 )θ2 (q 5/2 ) − θ2 (q 1/2 )θ23 (q 5/2 )
θ33 θ3 (q 5 ) − θ3 θ33 (q 5 )
φ(−q)φ 3 (−q 5 ) − φ 3 (−q)φ(−q 5 )
θ4 θ43 (q 5 ) − θ43 θ4 (q 5 )
16 qψ 3 (−q)ψ(−q 5 ) + q 2 ψ(−q)ψ 3 (−q 5 )
θ53 (q 1/2 )θ5 (q 5/2 ) + θ5 (q 1/2 )θ53 (q 5/2 )
−4
16
16
4
4
∞ χ
n
10b nq
n
1−q
1
∞ χ
1
∞
16
4(1 − 21−s − 22−2s ))C
−4(1 + 22−s )C
16(1 − 2−s )(1 + 22−s )C
n
10a q
(1 − q n )2
16(1 + 2−s )D
χ 10b q n
[(1 + (−q)n ]2
χ 5qn
(1 + q n )2
1
∞ χ
1
16(1 − 2−s )C
n
10b nq
2n
1−q
1
∞ χ
1
∞
Mellin transform
n
10a q
n
(1 + q )2
4(1 + 21−s − 22−2s ))D
4(1 − 22−s )D
16(1 + 2−s )(1 − 22−s )D
Table A.5 Modular equations of order 7, and mixed equations of orders 3, 5, and 15
Ramanujan
4qψ(q)ψ(q 7 )
φ(q)φ(q 7 ) − 1
Classical
θ2 (q 1/2 )θ2 (q 7/2 )
θ3 θ3 (q 7 ) − 1
Lambert series
4
2
n
−14a q
1 − qn
1
∞ χ
n
−14b q
1 + (−q)n
1
φ(−q)φ(−q 7 ) − 1
Mellin transform
∞ χ
∞ χ
n
−7 q
1 + qn
θ4 θ4 (q 7 ) − 1
−2
θ5 (q 1/2 )θ5 (q 7/2 )
∞ χ
n
−14a q
4
n
1+q
4(1 − 2−s )L 1 L −7
2(1 − 21−s + 21−2s )L 1 L −7
−2(1 − 21−s )L 1 L −7
1
4qψ(−q)ψ(−q 7 )
4 q 2 ψ(q)ψ(q 15 ) + qψ(q 3 )ψ(q 5 )
φ(q)φ(q 15 ) + φ(q 3 )φ(q 5 ) − 2
θ2 (q 1/2 )θ2 (q 15/2 ) + θ2 (q 3/2 )θ2 (q 5/2 )
θ3 θ3 (q 15 ) + θ3 (q 3 )θ3 (q 5 ) − 2
4
2
0
∞ χ
−30a q
1 − qn
1
∞ χ
θ4 θ4 (q 15 ) + θ4 (q 3 )θ4 (q 5 ) − 2
−2
−30b q
n
1 + (−q)n
1
φ(−q)φ(−q 15 ) + φ(−q 3 )φ(−q 5 ) − 2
n
∞ χ
n
−15 q
1 + qn
4(1 − 2−s )(1 − 21−s )L 1 L −7
4(1 − 2−s )L 1 L −15
2(1 − 21−s + 21−2s ))L 1 L −15
−2(1 − 21−s )L 1 L −15
1
4 q 2 ψ(−q)ψ(−q −15 ) − qψ(−q 3 )ψ(−q 5 )
θ5 (q 1/2 )θ5 (q 15/2 ) − θ5 (q 3/2 )θ5 (q 5/2 )
4
∞ χ
n
−30a q
1 + qn
1
4(1 − 2−s )(1 − 21−s )L 1 L −15
A.2 Character table for Dirichlet L-series
Table A.6 Character table for the Dirichlet L-series (see (1.4.1)). The column headings are values of the integer n. The vertical bars
indicate the ends of number pattern sequences
n=
1
χ1
χ 2a
χ 2b
χ −3
χ 3a
χ −4
χ5
χ 6a
χ −6a
χ −6b
χ −7
χ 10a
χ 10b
χ 12
χ −14a
χ −14b
χ −15
χ −30a
χ −30b
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
0|
1
0|
1
0|
1
0| 1
0|
1
0|
1
0|
1
0|
1
0|
1
0|
1 −1|
1 −1|
1 −1|
1 −1| 1 −1|
1 −1|
1 −1|
1 −1|
1 −1|
1 −1|
1 −1 0|
1 −1
0|
1 −1 0|
1 −1
0|
1 −1 0|
1 −1
0|
1 −1
1
1 0|
1
1
0|
1
1 0|
1
1
0|
1
1 0|
1
1
0|
1
1
1
0 −1
0|
1
0 −1
0| 1
0 −1
0|
1
0 −1
0|
1
0 −1
0|
1 −1 −1
1 0|
1 −1 −1 1
0|
1 −1 −1
1 0|
1 −1 −1
1
0|
1
0
0
0
1
0|
1
0 0
0
1
0|
1
0
0
0
1
0|
1
0
1
0
0
0 −1
0|
1
0 0
0 −1
0|
1
0
0
0 −1
0|
1
0
1
1
0 −1 −1
0|
1
1 0 −1 −1
0|
1
1
0 −1 −1
0|
1
1
1
1 −1
1 −1 −1 0|
1 1 −1
1 −1 −1
0|
1
1 −1
1 −1 −1
1
0 −1
0
0
0 −1
0 1
0|
1
0 −1
0
0
0 −1
0
1
0|
1
1 −1 −1
0 −1 −1
1 1
0|
1
1 −1 −1
0 −1 −1
1
1
0|
1
0
0
0 −1
0 −1
0 0
0
1
0|
1
0
0
0 −1
0 −1
0
1
0 −1
0 −1
0
0
0 1
0
1
0 −1
0|
1
0 −1
0 −1
0
1 −1 −1 −1 −1
1
0 −1 1
1
1
1 −1
0|
1 −1 −1 −1 −1
1
1
1
0
1
0
0 −1
1 0
0 −1
0 −1 −1 0|
1
1
0
1
0
1
0
0
0
0
0 −1
0 0
0 −1
0 −1
0
0
0
1
0
1
0
1 −1
0 −1
0
0 −1 −1 0
0 −1
0 −1
1
0 −1
1
0
1
0
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20 21
22
23
24 25
1
1
1
1
1
0|
1
0|
1 −1|
1 −1|
0|
1 −1
0|
0|
1
1
0|
1
0 −1
0|
1 −1 −1
1
0
0
1
0|
0
0 −1
0|
0 −1 −1
0|
0|
1
1 −1
1
0 −1
0
1
1 −1 −1
0
0
1
0|
0
0
1
0
0 −1
1
1
0 −1
1
0
0
0
1
0
0
1
1
0
26
27
28
29
30
1
1
1
1
1
1
1
0|
1
0|
1
0|
1 −1|
1 −1|
1 −1|
1 −1 0|
1 −1
0|
1
1 0|
1
1
0|
1
0 −1
0|
1
0
0|
1 −1 −1
1
0|
1
0
0
0
1
0|
1
0
0
0 −1
0|
1
1
0 −1 −1
0|
1 −1 −1
0|
1
1
0
0 −1
0
1
0|
0 −1 −1
1
1
0|
1
0
0
0 −1
0
1
0 −1
0|
1
0
1
1 −1
0|
1 −1
0 −1
0 −1 −1
0|
0
0
0
0 −1
0|
0
1
0
1 −1
0|
A.3 Values of K [N ] for all integer N from 1 to 100
331
A.3 Values of K [N ] for all integer N from 1 to 100
The relevant definitions are to be found in Section 1.12.
1 1
1
2
, k1 = ;
b
4 4
2
√ 1/2
√
4+2 2
2
b 18 , k 2 = 2
2 − 1 , Q[2] = 2;
K [2] =
16
√
√
2+ 3
22/3 3 1 2
K [3] =
b 3 , k3=
;
12
4
√
√
2
2 + 2 1
2
b 4 , k 4 = 25/2
2−1 ;
K [4] =
16
1/2
√
√ 1/4
3/5
5
−
2
1
+
2
2 × 53/8 2 + 5
1
, k52 =
K [5] =
b 20
,
40
2
W [5] = 3;
√
√ 1/2
√
27/12 31/4 1 + 2 + 6
√ 2 √ 2
1
, k62 =
2− 3 ,
b 24
3− 2
K [6] =
48
√
2
P[6] =
2−1
√ ⎡b 1 b 2 ⎤
√
7
7⎣ 7
8−3 7
2
⎦
, k7 =
;
K [7] =
14
16
b 3
K [1] =
7
K [8] =
√
√ 1/2
4+6 2+8 1+ 2
1
8 ,
32
√
4 √ 1
1+4 2 ;
k82 + 2 = 2
2+1
k8
2
√
√
√ √
2− 3
3+1
2 − 31/4
31/4 2
K [9] =
b 14 , k92 =
,
24
2
√ 4
V [9] = 2 − 3 ;
√
√
√ 1/2 ⎡ 1 9 ⎤
21/2 5 2 + 6 5 + 2 10
b 40 b 40
⎦,
⎣
K [10] =
80
b 18
√
2 √
4
2
=
10 − 3
2 − 1 , Q[10] = 3;
k10
b
332
Appendix
√ 1/3 √ 1/3 1/2
661/2 4 + 19 + 3 33
+ 19 − 3 33
K [11] =
132
⎡ ⎤1/3
1
3
4
5
b 11
b 11
b 11
b 11
⎦ ;
⎣
×
2
b 11
√
√ √
√
4
6+ 2+4
22/3 3
√ 4 √
2
=
3− 2
2−1 ;
K [12] =
b 13 , k12
96
√
1/4 ⎡ ⎤
1
9
b 52
133/8 × 210/13 5 13 + 18
b 52
⎦,
⎣
K [13] =
3
104
b 52
√
1 − 6 5 13 − 18
2
, W [13] = 11;
k13 =
2
√
√
29/14 × 75/8 √
2 + 1 3 + 6 2 + 2 14
K [14] =
112
⎡ ⎤1/2
'
1/4 b 5 b 1 b 13
√
√
56
8
56
⎣
⎦ ,
+ 2 2 + 12 2 + 4 14
17
b 56
√
Q[14] = 1 + 2 2;
√ 1/2 ⎡ 1 4 ⎤
b 15 b 15
61/2 5 + 5
⎦,
⎣
K [15] =
5
60
b 15
2
k15
=
√
K [16] =
√ 2 √
√ 2
√ 2 2− 3
3− 5
5− 3
2 + 21/4
32
128
2
b
1
4
,
√
2
k16
= %
2−1
;
4
21/4 + 1
&8 ;
1/4
1/2
√
√
221/34 177/16
5
17 + 4
+ 6 + 2 17
K [17] =
136
⎡ ⎤1/2
1
9
b 68
b 68
b 13
68
⎣
⎦ ,
×
15
b 68
√
17 + 5 17
W [17] =
;
2
A.3 Values of K [N ] for all integer N from 1 to 100
K [18] =
√
√ 1/2
28 + 10 2 + 8 3
b
1
8
,
6
√ 4 √
2
k18
= 2− 3
2−1 ,
48
√
√ 4
P[18] =
3− 2 ;
√ 1/3 √ 1/3 1/2
1/2
4 + 163 + 9 57
114
+ 163 − 9 57
K [19] =
228
⎡ % & ⎤1/3
1
4
5
6
7
9
b 19 b 19 b 19 b 19
b 19
b 19
⎦ ;
×⎣
2
3
8
b 19
b 19
b 19
1/2
√
5−2
1+2
1 + k5
K [20] =
K [5],
k52 =
,
2
2
( √
3
√ √ 1
2
5+2
k20 + 2 = 2
3 62 + 11 5 + 32 5 2 + 5 5 ;
k20
11/21
√ √ √ 1/4
213/8 2
2 2+ 3 3+ 7 + 7
K [21] =
168
⎡ ⎤1/2
1
5
b 84
b 17
b 84
84
⎦ ,
× ⎣
b 14
√ 2
√ 2 √
3 3−2 7 ;
V [21] = 8 − 3 7
√
√
√ 1/2 ⎡ 1 9 19 ⎤
b 88 b 88 b 88
239/44 33 2 + 7 11 + 5 22
⎦,
⎣
% &
K [22] =
5
7
176
b 88
b 88
Q[22] = 6;
√ √ 1/3 √ 1/3 1/6
65/6 23
K [23] =
20 + 6308 + 84 69
+ 6308 − 84 69
276
⎡ ⎤1/3
1
2
3
4
6
8
9
b 23
b 23
b 23
b 23
b 23
b 23
b 23
⎦ ,
% & ×⎣
5
7
11
b 23
b
b 23
b 10
23
23
⎡
1/3
1/3 ⎤
√
√
231 69 − 1181
231 69 + 1181
1 ⎣
⎦;
−22 − 5
+5
V [23] =
192
2
2
K [24] =
1+
√
3−1
√
√
√ 1/2
2+1
3− 2
2
K [6],
333
334
Appendix
√
√
√
√ 1/2
1−u 2
=
, u=
2+1
3+1
3− 2
,
1+u
√ √
√ 4 √ 1
2
209 + 104 3 + 48 2 4 + 3 ;
+ 2 =2 2+ 3
k24
k24
2 %
√
√
&2
3 − 2 × 51/4
5+2
5−2
2
b 14 , k25
,
K [25] =
=
20
2
W [25] = 47;
⎡ ⎤1/3
1
3
9
17
25
R 1/12 × 2233/312 × 135/12 ⎣ b 104 b 104 b 104 b 104 b 104 ⎦
,
K [26] =
23
208T [26]1/8
b 18 b 104
√ 2
√
√
1
R=
34 − 13 13 − 2
3 + 2 (4034
3
√ 2
√
√ 1/3 √
√
2
3− 2
+ 3100 6 − 2105 13 − 457 78
√
√
√ 1/3
× −4034 + 3100 6 + 2105 13 − 457 78
,
2
k24
√ 1/3 √ 1/3
1
8 + 359 − 12 78
;
+ 359 + 12 78
3
&2 % 2/3
2 +2
1
K [27] =
b 16 , V [27] = + 25 × 21/3 − 20 × 22/3 ;
72
4
√
√
8 + 3 2 + 14
K [7];
K [28] =
16
255/87 × 31/4 × 2911/24
K [29] =
696
⎧
⎤1/3 ⎡ √
⎤1/3 ⎫
⎡ √
⎪
⎪
⎬
⎨
2
2
87 + 9
87 − 9
⎣
⎣
⎦
⎦
−
× −1 +
⎪
⎪
9
9
⎭
⎩
⎧
( 1/3
⎨ 9 + √29
√ √
×
+ 369 + 70 29 + 12 6 27 + 5 29
⎩
2
⎫
( 1/3 ⎬3/4
√
√ + 369 + 70 29 − 12 6 27 + 5 29
⎭
Q[26] =
⎡ ⎤1/3
1
5
9
13
25
b 116
b 116
b 116
b 116
b 116
⎦ ,
× ⎣
% 7 &
23
b 116
b 116
√ 1/3 √ 1/3
1
W [29] =
71 + 343601 − 2688 87
;
+ 343601 + 2688 87
3
A.3 Values of K [N ] for all integer N from 1 to 100
335
√
√
√
√
√ 1/2
241/60 × 31/4 1/2 4 + 2 + 2 3 + 5 + 3 6 + 10
5
240
⎡ ⎤1/2
1
11
17
b 120
b 120
b 120
⎦ ,
× ⎣
% 7 &
b 120
√
√
P[30] = ( 10 − 3)2 ( 5 − 2)2 ;
√
31 1/3
K [31] =
R
62
⎡ % & ⎤1/3
1
2
4
5
7
8
9
14
b 31
b 31
b 31
b 31
b 31
b 31
b 10
b 31
31 b 31
⎦ ,
× ⎣
3
6
12
13
15
b 31
b 11
b
b
b
b 31
31
31
31
31
⎡
√ 1/3 ⎤
√ 1/3 187 + 9 93
187 − 9 93
1
⎦;
+
R = ⎣5 +
3
2
2
K [30] =
1 + k8
K [8];
2
37/66
√
√
√
√
× 333/8
2
(3 + 11)1/2 ( 3 + 1)3/2 (2 3 − 11)1/4
K [33] =
528
⎡ ⎤1/2
1
17
25
29
b 132
b 132
b 132
b 132
⎦ ,
× ⎣
31
b 14
b 132
√
√
V [33] = (2 − 3)6 (10 − 3 11)2 ;
√
2123/136 × 173/8 9 − 4 33 17 − 131
K [34] =
272
T [34]1/8
⎤1/2
⎡
1
9
19
25
33
b 136
b 136
b 136
b 136
b 136
⎦ ,
⎣
×
13
15
21
b 136
b 136
b 136
√
Q[34] = 3/2( 17 + 3);
√
√
√
√
1
K [35] =
2 + 2 5 + (146 + 91 5 − 3 21 + 6 105)1/3
2
1/2
√
√
√
+(146 + 91 5 + 3 21 + 6 105)1/3
⎡ ⎤1/3
1
3
13
16
b 35
b 35
b 11
35 b 35 b 35
⎦ ;
×⎣
2
b 15
b 35
√ √
√
31/4 2 1 + 2 + 3 + 31/4
b 14 ;
K [36] =
48
K [32] =
336
Appendix
⎡ ⎤
1
9
21
25
33
√ 1/2 b 148 b 148 b 148 b 148 b 148
22/37 × 373/8
⎦,
% & (6 + 37) ⎣
K [37] =
3
7
11
27
148
b 148
b 148
b 148
b 148
W [37] = 146;
R 1/12 × 2347/456 × 195/12
304T [38]1/8
⎡ ⎤1/3
1
3
9
17
25
27
b 152
b 152
b 152
b 152
b 152
b 152
⎦
× ⎣
5
15
31
b 152
b 152
b 152
√
√
√
√
( 2 + 1)3
2(2747 + 859 2)
R=−
11 + 18 2 +
+ 2(7 − 5 2)S ,
3
S
√ 1/3
√ √
√
,
S = (358573 + 368494 2)( 2 + 1)4 + 66 57(32771 + 23166 2)
√
√
1/3
1/3
1
+ 1369 + 30 114
Q[38] =
16 + 1369 − 30 114
;
3
1/4
'
√
√
√
⎡ ⎤
2
5
13 108 39 + 11 13 + 6(325 + 91 13
b 39
b 39
⎦;
⎣
K [39] =
%7&
468
b 39
K [38] =
1 + k10
K [10];
2
'
1/8
√
271/82 × 4115/32
R 1/8
K [41] =
−8 + 32 + 5 41
328
⎡ ⎤1/4
1
5
9
21
25
33
37
b 164
b 164
b 164
b 164
b 164
b 164
b 164
⎦ ,
×⎣
23
31
39
b 164
b 164
b 164
1
v + v2 − 4 ,
R=
4 √
v = 5 41 + 32
'
√ √
√
× 33440 + 5162 41 + 32 211 + 35 41 × 10 + 2 41 ,
'
√
√ 1
W [41] = 207 + 31 41 + 2(42281 + 6605 41) ;
4
√
√
√
√
√
231/84 × 33/4 × 71/4
K [42] =
(6 + 4 2 + 2 3 + 5 6 + 7 + 42)1/2
1008
⎡ ⎤
1
25
b 168
b 168
⎣
⎦,
×
37
b 168
√ 3
√
√
5 − 21
(2 2 − 7)2 ;
P[42] =
2
K [40] =
A.3 Values of K [N ] for all integer N from 1 to 100
'
K [43] =
337
√
√
258(16 + (3547 + 276 129)1/3 + (3547 − 27 129)1/3 )
516
⎤1/3
1
4
6
9
11
13
14
b 43
b 43
b 43
b 43
b 10
43 b 43 b 43 b 43
⎢
⎥
⎢
⎥
16
17
21
⎢
⎥
b
b
b
× b 15
43
43
43
43
⎢
⎥
× ⎢ % & ⎥
⎢ b 2 b 3 b 5 b 7 b 8 b 12 b 18 b 19 b 20 ⎥
⎢ 43
43
43
43
43
43
43
43
43 ⎥
⎣
⎦
⎡
1 + k11
K [11]
2
√
√
1+2 3+2 5
K [45] =
K [5]
3
√
√
1/8
2177/184 × 233/8 9(6 2 − 7) − 14 36 2 − 50
K [46] =
368
T [46]
⎡ ⎤1/2
1
9
11
19
25
41
43
b 184
b 184
b 184
b 184
b 184
b 184
b 184
⎦ ,
% & ×⎣
7
13
15
29
b 184
b 184
b 184
b 18 b 184
√
Q[46] = 3( 2 + 1)2 ;
√
47
[a(a + 1)]2/5
K [47] =
94
⎡ % & ⎤1/5
1
2
3
4
6
7
8
9
b 47
b 47
b 47
b 47
b 47
b 47
b 47
b 12
b 47
47 ⎥
⎢
⎢
⎥
14
16
17
18
21
⎢
⎥
× b 47 b 47 b 47 b 47 b 47
⎢
⎥
× ⎢ ⎥
⎢ b 5 b 10 b 11 b 13 b 15 b 19 b 20 b 22 b 23 ⎥
⎢ 47
47
47
47
47
47
47
47
47 ⎥
⎣
⎦
K [44] =
where a is the only real root of the equation a 5 − a 3 − 2a 2 − 2a − 1 = 0:
1,
s(1)1/5 + s(2)1/5 − [−s(3)]1/5 + s(4)1/5 ,
a=
5
√
s(n) = Re −650 + 80 −47) exp − 25 πin
√
+ −975 + 15 −47 exp − 45 πin ;
√
√
√ √
√ 3/2
2−1
3− 2
6+ 2
1+
K [12];
K [48] =
2
338
Appendix
K [49] =
√ √
√ 1/2
7 + 4 7 + 71/4 2 5 + 7
b
28 √
√ W [49] = 8 + 3 7 16 + 3 7 ;
1
4
,
√
√
√
√
√
K [50] = 5 − 2 5 + 2 10 + 2 55 − 30 2 − 11 5 + 10 10
(
√
√
√ + 3 30 29 − 20 2 − 11 5 + 8 10
1/3
√
√
√
+ 2 55 − 30 2 − 11 5 + 10 10
(
√
√
√ − 3 30 29 − 20 2 − 11 5 + 8 10
⎫
1/3 ⎬
⎭
×
K [2]
,
15
√ 1/3 √ 1/3
1
16 + 5 2279 − 84 6
;
+ 5 2279 + 84 6
3
√
√ √ 1/3 √ 1/3 1/2
35/6 102 K [51] =
2 1 + 17 + 497 + 151 17
+ 623 + 151 17
612
⎡ ⎤1/3
1
5
13
16
20
b 51
b 11
b 17
51 b 51 b 51 b 51
⎦ ;
% & ×⎣
2
7
8
b 51
b 51
b 51
Q[50] =
K [52] =
K [53] =
1 + k13
K [13];
2
1/3
√
1/3 1/3
√
− 2 3 159 − 10
−1 + 2 3 159 + 10
219/106 × 47711/24
636
R 1/4
⎡ ⎤1/3
1
9
13
17
25
29
37
49
b 212
b 212
b 212
b 212
b 212
b 212
b 212
b 212
⎦ ,
×⎣
% 7 &
11
15
43
47
b 212
b 212
b 212
b 212
b 212
√
√
R=
53 − 7 1661 − 217 53
√ 1/3
√
+ 16 2 2077192 + 165 3 − 285355 53
√
√ 1/3
− 16 2 −2077192 + 165 3 + 285355 53
,
√ 1/3
1
505 + 8 266489 − 210 159
W [53] =
3
√ 1/3
;
+ 8 266489 + 210 159
A.3 Values of K [N ] for all integer N from 1 to 100
339
√
2 √
√ 1/3
√
2+1
9−5 3
K [54] =
2−1
3− 6+2 6
9
√
√ 1/3
√
+ 6
6+6 3−9 2
K [6];
K [55] =
'
√ 1/4
23/4 × 57/8 × 111/4 √
5+1 3+ 3+2 5
440
% & 2
7
b 17
b 55 b 55
55
;
×
3
23
b 55 b 55
1 + k14
K [14];
2
3/2 √ 1/4
√ 1/2 √
225/38 × 573/8 √
13 + 3 19
2 19 − 5 3
3+1
K [57] =
912
⎡ ⎤1/2
1
25
29
41
49
53
b 228
b 228
b 228
b 228
b 228
b 228
⎣
⎦ ,
×
% 7 & 43 55 1 b 228 b 228 b 4
b 228
√ 6 √ 2
170 − 39 19 ;
V [57] = 2 − 3
K [56] =
K [58] =
√
√ 1/2
257/58 × 581/4 70 + 99 2 + 13 29
464
⎡ ⎤
1
9
25
33
35
49
51
57
b 232
b 232
b 232
b 232
b 232
b 232
b 232
b 232
⎦,
% & × ⎣
5
7
13
23
45
53
b 232
b 232
b 232
b 18 b 232
b 232
b 232
Q[58] = 27;
K [59] =
√ 1/3
√
√ 1/3 1/3 2
62/3 59
+ 22/3 43 + 3 177
R
−8 + 22/3 43 − 3 177
1416
⎡ % & ⎤1/9
1
3
4
5
7
9
15
16
b 59
b 59
b 59
b 59
b 12
b 59
b 59
59 b 59 b 59
⎦
×⎣
2
6
8
11
b 59
b 59
b 59
b 10
b
59
59
⎡ ⎤1/9
17
19
20
21
22
25
27
28
29
b 59 b 59 b 59 b 59 b 59 b 59 b 26
59 b 59 b 59 b 59
⎦ ,
×⎣
14
18
23
24
b 13
59 b 59 b 59 b 59 b 59
R = α + (β + γ )1/3 + (β − γ )1/3 ,
1/3
1/3 √
√
1
4 + 22/3 43 − 3 177
,
+ 22/3 43 + 3 177
α=
9
1
129 + 3 × 22/3 (u − v) − 2 × 21/3 u 2 + v 2 ,
β=
243
340
Appendix
√
1/3
√
1/3
u= 3
177 − 9
, v= 3
177 + 9
,
√ 1/3 1/2
√ 1/3
1
− 4 44 − 3 177
;
25 − 4 44 + 3 177
γ =
9
√ √ √ √
√ 5+ 3
16 + 2 3 − 5 2 + 3
K [60] =
K [15];
32
2139/183 × 32/3 × 6111/24
K [61] =
1464
√
1/3
√
1/3 1/3
−5 + 2 6 183 + 62
− 2 6 183 − 62
×
V [61]1/8
⎡ ⎤1/3
1
5
9
13
25
41
b 244
b 244
b 244
b 244
b 244
b 244
⎢
⎥
⎢
⎥
45
49
57
⎢
⎥
b 244
b 244
× b 244
⎢
⎥
×⎢ ⎥ ,
⎢ b 3 b 15 b 19 b 27 b 39 b 47 ⎥
⎢
244
244
244
244
244
244 ⎥
⎣
⎦
√ 1/3
√ 1/3
;
+ 3 976247 + 144 183
W [61] = 296 + 3 976247−144 183
K [62] =
1/8
2189/248 × 317/16
R
496
T [62]
⎡ ⎤1/4
1
3
9
11
25
27
33
b 248 b 248 b 248
b 248
b 248
b 248
b 248
⎢
⎥
⎢
⎥
41
43
49
⎢
⎥
b 248
b 248
× b 248
⎢
⎥
×⎢
⎥ ,
⎢
⎥
5
15
23
1
45
55
b 248 b 248 b 248 b 8 b 248 b 248
⎢
⎥
⎣
⎦
'
1/2
'
√
√
√
√
,
R = 25 − 19 2 + 2 2(−13 + 10 2 − 10 2−16 2 + 8 1 + 4 2
√
Q[62] =
√
K [63] =
K [64] =
2+1
2
'
2
√ 1/2
3 + 9 + 2 21
5+
√
112 2 − 127 ;
K [7];
√
1/2
√ % 1/4
&2
2 2 + 1 + 217/8
2+1
6
64
b
1
4
;
A.3 Values of K [N ] for all integer N from 1 to 100
K [65] =
2
√
46/65
65
341
( √ 1/4
√ 2 1 + 65
−64 + 7 + 65
1040
V [65]1/8
⎡ ⎤1/4
1
9
29
33
37
49
57
61
b 260
b 260
b 260
b 260
b 260
b 260
b 260
b 260
⎦ ,
⎣
×
% 7 & 47 51 63 b 260
b 260 b 260 b 260
√ 1/2
√
;
W [65] = 291 + 36 65 + 8 5 524 + 65 65
√
1/8
R
25/88 × 35/8 11
K [66] =
264
T [66]
⎡ ⎤1/2
1
17
9
41
65
b 264
b 88
b 264
b 19
b 88
88 b 264
⎦ ,
% & ×⎣
5
7
29
31
b 264
b 88
b 264
b 88
√
√
√ √
R = 27 − 5 6 − 3 22 − 2
3− 2
√ √
√ 1/2
√
× −29 − 2 6 + 11 11 3 − 6 2
,
√
√
3
P[66] = u − u 2 − 1, u = 3 2 3 + 11
'
1/2
√
√
√
× 15 11 − 22 3 + 4 −26 + 6 33
.
Note the b’s that appear above with 88 in their denominator are precisely those that
appear in K [22], leading to the following remarkable result:
3R 1/4
1
√
√
√
T [66]
(33 2 + 7 11 + 5 22)1/2
⎡
⎤
17
41
65
b 264
b 264
b 264
⎦.
× ⎣
29
31
b 264
b 264
25/22
K [66]2
=
K [22]
24
√ √ 1/3
√ 1/3 67
+ 710 − 18 201
K [67] =
8 + 710 − 18 201
804
⎡ ⎤1/3
1
4
6
9
14
15
16
17
b 67
b 67
b 67
b 10
b 67
67 b 67 b 67 b 67 b 67
⎦
% & ×⎣
3
5
7
8
12
13
2
b 67
b 67
b 67
b 67
b 11
b
b
b 67
67
67
67
⎡ ⎤1/3
21
22
23
24
25
26
29
33
b 19
67 b 67 b 67 b 67 b 67 b 67 b 67 b 67 b 67
⎦ ;
⎣
×
20
27
28
30
31
32
b
b
b
b
b
b
b 18
67
67
67
67
67
67
67
342
Appendix
K [68] =
K [69] =
1 + k17
K [17];
2
√ 1/8
R
2173/276 × 35/16 23
552
V [69]
⎡ % & ⎤1/4
1
5
7
13
25
35
49
65
b 276
b 276
b 276
b 276
b 276
b 276
b 276
b 276
⎦ ,
⎣
×
1
29
37
61
b 276
b 276
b 276
b 92
'
√
√ √
R = 22 − 9 3 + 12 − 8 3
5 + 4 3,
1
V [69]
'
√
√
√
= 2(2 + 3)7 36002 + 27401 3 + 288 138(302 + 177 3) ;
V [69] +
( √
√
√
√ √
263/140 × 51/4 × 71/2
K [70] =
35 11 + 5 2 + 6 5 + 6 7 + 3 10 + 3 70
19600
⎡ ⎤
1
9
19
59
b 280
b 280
b 280
b 280
√
⎦ , Q[70] = 3(7 + 2 10);
⎣
×
29
31
b 280
b 18
b 280
√ 2/7
71 2
a (a + 1)
K [71] =
142
⎡ ⎤1/7
1
2
3
4
5
6
8
9
b 71
b 71
b 71
b 71
b 71
b 71
b 71
b 71
⎢
⎥
⎢
⎥
12
15
⎢
⎥
b
b
× b 10
71
71
71
⎢
⎥
×⎢
⎥
%
&
⎢
⎥
7
11
13
14
17
21
22
b 71 b 71 b 71 b 71 b 71 b 71 b 71
⎢
⎥
⎣
⎦
⎡ ⎤1/7
18
19
20
24
25
27
b 16
71 b 71 b 71 b 71 b 71 b 71 b 71 ⎥
⎢
⎢
⎥
30
32
⎢
⎥
× b 29
71 b 71 b 71
⎢
⎥
×⎢ ⎥ ,
⎢ b 23 b 26 b 28 b 31 b 33 b 34 b 35 ⎥
⎢
71
71
71
71
71
71
71 ⎥
⎣
⎦
where a is the only real root of the equation
a 7 − 2a 6 − a 5 + a 4 + a 3 + a 2 − a − 1 = 0,
z = exp − 27 iπ ,
A.3 Values of K [N ] for all integer N from 1 to 100
343
√
s(n) = Re − 13829 + 2849 −71 z n
√
√
− 43152 + 3752 −71 z 2n − 63690 + 1778 −71 z 3n ;
1,
a=
2 + s(1)1/7 + s(2)1/7 − [−s(3)]1/7 + s(4)1/7 + s(5)1/7 + s(6)1/7 ,
7
√
√
√
√
1 + ( 3 + 2)3/2 (14 2 − 10 − 4 6)1/4
K [72] =
K [18];
2
1/8
R
265/73 × 737/16
K [73] =
584
V [73]
⎡
⎤1/2
1
9
25
37
41
49
57
b 292
b 292
b 292
b 292
b 292
b 292
b 292
⎢
⎥
⎢
⎥
61
65
69
⎢
⎥
b 292
b 292
× b 292
⎢
⎥
× ⎢ ⎥ ,
⎢ b 3 b 19 b 23 b 27 b 35 b 55 b 67 b 71 ⎥
⎢ 292
292
292
292
292
292
292
292 ⎥
⎣
⎦
√
1921 + 225 73
W [73] =
;
2
√
2227/296 × 379/20 (r1 / 37 + r2 )1/20
K [74] =
592
T [74]1/8
⎡ ⎤1/5
1
3
9
11
25
27
b 296
b 296
b 296
b 296
b 296
b 296
⎦
× ⎣
% 7 &
21
b 18
b 296
b 296
⎡ ⎤1/5
33
41
49
65
67
73
b 296
b 296
b 296
b 296
b 296
b 296
⎣
⎦ ,
×
47
53
63
71
b 296
b 296
b 296
b 296
√
R = 125 73 − 1068,
11a 4 + 18a 3 + 43a 2 + 56a + 52
,
2
r1 = 29369a 4 − 24094a 3 + 64167a 2 − 108496a + 37987,
Q[74] =
r2 = −204a 4 + 7954a 3 − 10850a 2 + 13232a − 4654,
1,
a = − 1 + s(1)1/5 + s(2)1/5 − [−s(3)]1/5 − [−s(4)]1/5 ,
5 √
√
s(n) = Re −2 + 100 −74 exp − 25 πin + 948 − 10 −74
× exp − 45 πin ;
√
1 K [75] =
5 + 2 × 101/3 + (5 + 3 5)10−1/3 K [3];
15
344
Appendix
K [76] =
1 + k19
K [19];
2
√
√ 1/4
√
√
11
−
5
13
+
4
11
+
143
+
44
11
2
+
2
11
K [77] =
1/8
616
V [77]
⎡ ⎤1/4
1
3
9
13
27
37
39
b 308
b 308
b 308
b 308
b 308
b 308
b 308
⎢
⎥
⎢
⎥
47
53
73
⎢
⎥
b 308
b 308
× b 308
⎢
⎥
×⎢
⎥ ,
⎢
⎥
5
1
15
23
67
69
b 308 b 44 b 308 b 308 b 308 b 308
⎢
⎥
⎣
⎦
2327/616
× 77/16
√
√
1229 + 368 11 + 35 2441 + 736 11
;
W [77] =
2
√
√
√
√
√ 1/2
291/156 × 33/4 × 131/4 10 + 15 2 + 12 3 + 17 6 + 3 13 + 4 26
K [78] =
1872
⎡ ⎤
1
25
35
49
b 312
b 312
b 312
b 312
⎦,
×⎣
1
23
61
b 312
b 312
b 24
√ 6
√
3 − 13
P[78] =
( 26 − 5)2 ;
2
√ 2/5
79
a3
K [79] =
158 (a − 1)
⎡ ⎤1/5
1
2
4
5
8
9
11
b 79
b 79
b 79
b 79
b 79
b 79
b 10
79 b 79 ⎥
⎢
⎢
⎥
16
18
⎢
⎥
× b 13
79 b 79 b 79
⎢
⎥
×⎢ % & ⎥
⎢ b 3 b 6 b 7 b 12 b 14 b 15 b 17 b 24 ⎥
⎢
79
79
79
79
79
79
79
79 ⎥
⎣
⎦
⎡
⎤1/5
b 20
b 21
b 22
b 23
b 25
b 26
b 31
79
79
79
79
79
79
79
⎢
⎥
⎢
⎥
36
38
⎢
⎥
b
b
× b 32
79
79
79
⎢
⎥
× ⎢ ⎥ ,
⎢ b 27 b 28 b 29 b 30 b 33 b 34 b 35 b 37 b 39 ⎥
⎢ 79
79
79
79
79
79
79
79
79 ⎥
⎣
⎦
b
19
79
where a is the only real root of the equation
a 5 − 3a 4 + 2a 3 − a 2 + a − 1 = 0,
A.3 Values of K [N ] for all integer N from 1 to 100
345
1,
3 + s(1)1/5 − [−s(2)]1/5 + s(3)1/5 + s(4)1/5 ,
5
√
s(n) = Re −(331 + 15 −79) exp(− 25 πin) − 616 exp(− 45 πin) ;
a=
K [80] =
1 + k20
K [20];
2
√ 1/3 2
1 + 2(1 + 3)
K [81] =
K [82] =
36
;
√ 1/2
√
−9 + 41 + −6 + 2 41
b
1
4
2183/328 × 413/8
656
T [82]1/8
⎡ ⎤1/2
1
9
25
33
43
49
51
b 328
b 328
b 328
b 328
b 328
b 328
b 328
⎢
⎥
⎢
⎥
57
59
73
81
⎢
⎥
b 328
b 328
b 328
× b 328
⎢
⎥
×⎢
⎥ ,
⎢
⎥
5
21
23
31
37
39
b
b
b
b
b
b
⎢
⎥
328
328
328
328
328
328
% &
⎣
⎦
45
61
77
× b 328 b 328 b 328
√
3
(19 + 3 41);
2
√
1/3
√
188/9 × 83 √
(5 249 + 9)1/3 − (5 249 − 9)1/3
R2
K [83] =
5976
⎡ % & ⎤1/9
1
3
4
7
9
11
12
b 83
b 83
b 83
b 83
b 10
b 83
83 b 83 b 83 ⎥
⎢
⎢
⎥
17
21
23
25
⎢
⎥
× b 16
83 b 83 b 83 b 83 b 83
⎢
⎥
× ⎢ ⎥
⎢ b 2 b 5 b 6 b 8 b 13 b 14 b 15 b 18 ⎥
⎢ 83
83
83
83
83
83
83
83 ⎥
⎣
⎦
Q[82] =
⎡ ⎤1/9
27
28
29
30
31
33
36
b 26
83 b 83 b 83 b 83 b 83 b 83 b 83 b 83 ⎥
⎢
⎢
⎥
38
40
41
⎢
⎥
× b 37
83 b 83 b 83 b 83
⎢
⎥
×⎢ ⎥ ,
⎢ b 19 b 20 b 22 b 24 b 32 b 34 b 35 b 39 ⎥
⎢
83
83
83
83
83
83
83
83 ⎥
⎣
⎦
2
+ (β + γ )1/3 + (β − γ )1/3 ,
3 √ 1/3
√ 1/3
1
+ 3 × 22/3 79 + 3 249
β=
29 + 3 × 22/3 79 − 3 249
,
27
√ 1/3 1/2
√ 1/3
1
+ 4 1432 + 3 249
;
49 + 4 1432 − 3 249
γ =
9
R=
346
Appendix
K [84] =
1 + k21
K [21];
2
√
√
√
√ 1/4
213/170 × 85 954 + 392 5 + 216 17 + 105 85
340
⎡ ⎤1/2
1
9
21
37
49
57
69
73
81
b 340
b 340
b 340
b 340
b 340
b 340
b 340
b 340
b 340
⎦ ,
% & ×⎣
3
7
19
23
27
59
63
b 340
b 340
b 340
b 340
b 340
b 340
b 340
√
3901 + 945 17
W [85] =
;
2
K [85] =
√
21341/1720 × 439/20 (r1 / 2 + r2 )1/20
K [86] =
592
T [86]1/8
⎡ ⎤1/5
1
3
9
17
19
25
b 344
b 344
b 344
b 344
b 344
b 344
⎣
⎦
×
% 7 &
13
21
39
b 344
b 344
b 344
b 344
⎡ ⎤1/5
27
41
49
51
57
75
81
b 344
b 344
b 344
b 344
b 344
b 344
b 344
⎦ ,
×⎣
53
55
63
71
b 344
b 344
b 344
b 344
3a 4 + 10a 3 + 13a 2 + 4a − 2
,
2
r1 = −2192a 4 − 29128a 3 − 59308a 2 + 94876a + 196322,
Q[86] =
r2 = −1563a 4 − 21058a 3 − 41223a 2 + 67132a + 139169,
1,
−1 + s(1)1/5 + s(2)1/5 − [−s(3)]1/5 + s(4)1/5 ,
a=
5
√
s(n) = Re −1158 + 20 −86 exp − 25 πin
√
− 1288 + 50 −86 exp − 45 πin ;
√ 1/3 √ 1/3 1/3 √
√ 1/6
+ 23+2 29
87
2 + 23−2 29
32(5 + 29)
K [87] =
348
⎡ ⎤1/3
1
2
8
13
17
22
25
26
b 87 b 87 b 87 b 87 b 14
87 b 87 b 87 b 87 b 87
⎦ ;
⎣
×
1
5
23
38
40
b 87
b 19
b
b
b
b 29
87
87
87
87
K [88] =
1 + k22
K [22];
2
A.3 Values of K [N ] for all integer N from 1 to 100
347
2167/267 × 8923/48 R 1/24
712
V [89]1/8
⎡ ⎤1/6
1
5
9
17
21
25
45
b 356
b 356
b 356
b 356
b 356
b 356
b 356
⎣
⎦
×
11
39
47
55
b 356
b 356
b 356
b 356
⎡ ⎤1/6
49
53
57
69
73
81
85
b 356
b 356
b 356
b 356
b 356
b 356
b 356
⎦ ,
×⎣
67
71
79
87
b 356
b 356
b 356
b 356
√
15649
1
(u − v)
R=
−15580 + 7583 89 + 4 1591 − √
3
89
81
+ 108 √ − 23 (u 2 + v 2 ) ,
89
√
√
u = (−91 + 6 267)1/3 ,
v = (91 + 6 267)1/3 ,
√
√
1 267(4879 + 519 89) − 4(48683 + 5149 89)(u − v)
W [89] =
1602
√
+ 22(2314 + 245 89)(u 2 + v 2 ) ;
K [89] =
K [90] =
√
√
√ √
6( 2 − 1)( 5 − 2 + 1)
6
√
√
√
√
√
√
√ 1/2 −45 + 33 2 − 54 3 − 21 5 + 39 6 + 15 10 − 24 15 + 17 30
+
3
× K [10];
( √
13 − 3
75/6 26
K [91] =
R2
728
⎡ ⎤1/3
5
1
9
22
23
24
25
b 91
b 13
b 91
b 20
91 b 91 b 91 b 91 b 91 ⎥
⎢
⎢
⎥
33
36
⎢
⎥
× b 31
91 b 91 b 91
⎢
⎥
×⎢ ⎥ ,
⎢ b 2 b 3 b 8 b 15 b 17 b 27 b 32 b 40 ⎥
⎢
91
91
91
91
91
91
91
91 ⎥
⎣
⎦
√ 1/3 √ 1/3
√
√
1
+ 44 + 9 13 + 3 21
2 + 44 + 9 13 − 3 21
;
3
√
√ 1/3
√
K [92] = 24 + 61/2 48 − 4 23 + 5 × 22/3 −21 3 + 17 23
√
√ 1/3 1/2 K [23]
2/3
21 3 + 17 23
+5×2
;
48
R=
348
Appendix
√
√
√ 1/4
224/31 × 933/8 2532 + 1170 3 + 455 31 + 210 93
744
⎡ % & ⎤1/2
1
17
25
29
49
53
65
77
89
b 372
b 372
b 372
b 372
b 372
b 372
b 372
b 372
b 372
⎦ ;
× ⎣
% 7 & 11 19 23 67 83 1 b 372 b 372 b 372 b 372 b 372 b 372 b 4
√
√
√
V [93] = (1520 − 273 31)2 (45 3 − 14 31)2 ;
K [93] =
1/8
477/16
R
376
2T [94]
⎡ ⎤1/4
1
9
11
17
19
25
35
b 376
b 376
b 376
b 376
b 376
b 376
b 376
⎦
× ⎣
15
21
23
31
37
b 376
b 376
b 376
b 376
b 376
⎡ ⎤1/4
43
49
65
67
81
89
91
b 376
b 376
b 376
b 376
b 376
b 376
b 376
⎦ ,
×⎣
39
53
61
87
b 376
b 18 b 376
b 376
b 376
√
√
R = −26 − 32 2 + (23 − 5 2)u
'
√
√
√
1
+
2 + 2(7 − 4 2 + u) 17 + 5 2 + 13(1 − 2)u ,
2
'
'
√
√
√
1
39 + 30 2 + 3 345 + 244 2 ;
u = 9 + 8 2, Q[94] =
2
√
19/24
5/16
13/24
2
×5
( 5 − 1)
× 193/8 5/6 1/3
R S
K [95] =
3040
⎤1/2
⎡
1
6
16
26
36
b 95
b 11
b 95
95 b 95 b 95 b 95
⎦ ,
×⎣
1
29
34
b
b
b
b 14
95
5
95
95
'
√
√
√
R = 5 + ( 5 + 2) 2 5 − 1
1/2
' √
√
+ ( 5 + 2)1/2 6 + 2 5(2 5 − 1)
,
'
√
√
√
S = −(3 + 5) 2 5 − 1 + 5(2 5 − 1)
K [94] =
'
√ 1/2
√
−21/2 13 − 10 5 + 91 + 122 5
K [96] = (R −
R 2 − R)K [24],
R = (2 +
√
√ 2 √
3) ( 3 + 2)2 ;
;
A.3 Values of K [N ] for all integer N from 1 to 100
349
1/8
R
2187/194 × 977/16
776
V [97]
⎡ ⎤1/2
1
9
25
33
49
53
b 388
b 388
b 388
b 388
b 388
b 388
⎦
× ⎣
3
11
27
31
45
b 388
b 388
b 388
b 388
b 388
⎡ ⎤1/2
61
65
73
81
85
89
93
b 388
b 388
b 388
b 388
b 388
b 388
b 388
⎦ ,
⎣
×
43
47
75
79
91
95
b 388
b 388
b 388
b 388
b 388
b 388
√
√
7537 + 765 97
R = 569 97 − 5604, W [97] =
;
2
'
√
√
K [98] = M7 (2)K [2], Q[98] = 23 + 6 14 + 2 2(119 + 32 14).
K [97] =
One of the simplest equations for the seventh-order multiplier seems to be
7M7 (μ) −
1
= 6 − 16t + 12t 2 − 8t 3 ,
M7 (μ)
t 4 = kμ k49μ ,
and alternatives originating with Ramanujan are studied in Section 9.5 of the Borweins’ Pi and
the AGM.
√
√
[9 + 2(21 33 − 27)1/3 + 2(54 + 6 33)1/3 ]1/2
K [99] =
K [11];
27
√
√
4 + 2 5 + 2(3 + 2 × 51/4 ) 1 K [100] =
b 4 .
80
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Index
β, see beta function
(k, l)± , see Hurwitz zeta function
(k, l : s), see Hurwitz zeta function
(m|n), see Kronecker symbol
(s)n , see Pochhammer symbol
p Fq , see hypergeometric function
G, see Catalan’s constant
Iν (x), see modified Bessel function
Jν (x), see Bessel function
(k), 204
K (k), K [N ], see elliptic integral
K ν (x), see modified Bessel function
L-series, see Dirichlet L-series
L k (s), see Dirichlet L-series
Ms [ f (t)], see Mellin transform
q-series, 33, 65, 114, 198, 200, 206, 216, 217
Q(a, b, c; s), see two-dimensional lattice sum
Q 0 , Q 1 , Q 2 , Q 3 , see Jacobi’s products Q i
S(a, b, c; s), see two-dimensional lattice sum
S( p, r, j; s), see displaced lattice sum
T ( p 2 , 0, −r 2 ; s), see quadratic form, indefinite
W B , W F , W S , see Watson integrals
α(s), see Dirichlet eta function
β(s), see Dirichlet beta function
γ , see Euler–Mascheroni constant
(z), see gamma function
Z | hg |, see Epstein zeta function
ζ (s), see Riemann zeta function
ζ (s, q), see Hurwitz zeta function
η(τ ), see Dedekind eta function
η(s), see Dirichlet eta function
θ ( j, p), 164, 165
θ (k, l), 202
θ2 (q), θ3 (q), θ4 (q), see theta functions
, 12, 20, 28, 129, 130, 150, 179
φ(k, l), 202
φ(q), see Ramanujan notation
χ (k, l), 202
χk (n), see Dirichlet L-series
ψ(q), see Ramanujan notation
?
=, 243
Abel, Niels, 304, 313
Abel summable expression, 109, 117
Abelian group, 58, 218
absolute convergence, xiii, 3, 10, 87, 91, 101,
110, 150, 269
alternating series, see test, alternating series, 59,
97, 100, 112, 184, 222, 289
analytic continuation, 3, 39, 101, 103, 104, 110,
111, 113, 120, 178, 219, 222, 228–231,
236, 238, 257, 259, 290
Andrews, George, 116
angular-dependent sums, 138, 144, 148
anisotropic lattice, 306, 308
Appell, P., xvii, 1, 2, 9, 30, 301–303, 308, 312
asymptotics, 29, 43, 53, 149, 179, 297, 303
Bailey, D. H., 314
Bailey, W. N., 301, 302
Benson, G. C., xiii, 27, 29, 43
Berndt, B. C., 170, 171, 186
Bernoulli, Jacob
Bernoulli numbers, 181
Bernoulli polynomials, 158, 290
Bessel function, 17, 68, 135, 140, 253, 269, 270,
302, 316
addition theorem, 141
modified Bessel function, 6, 27, 43, 44, 68,
74, 132–135, 146, 149, 151, 222, 243,
314, 316
recurrence, 137, 143, 316
table for 2D sums, 44
beta function, 74, 75, 120, 180, 317
binomial theorem, 68
Bloch factor, 140
body-centred cubic (BCC) lattice, 64, 73, 228,
234, 236, 295
Born, Max, xvii, 1, 4, 14, 15, 20, 21, 42, 228
Index
Born’s Grundpotential, 15, 20
Borwein, D., xi, 229, 232, 259, 265, 285, 289,
290
Borwein, J. M., xi, 75, 192, 197, 229, 232, 259,
265, 285, 289, 290, 305, 349
Borwein, P. B., xi, 192, 197, 229, 349
Bravais lattice, 9, 10, 15, 20, 21, 23, 34, 235
Broadhurst, D., 76, 314
cardinal point, 146–148
Catalan, Eugène, 159
Catalan’s constant, 72, 138, 178, 239, 292
Cauchy, Augustin-Louis, 106, 110, 114
Cauchy principal value, 142
Cauchy residue theorem, 63, 290
central beta function, see beta function
Chaba, A. N., 30, 43, 47, 50, 52
character, see Dirichlet L-series
Chowla, S., 66, 177
class number, 55, 58, 76, 159, 221
Clausius, Rudolf, 126
closed form, 28, 30, 48, 63, 66, 73, 78, 116, 122,
128, 143, 146, 157, 164, 174, 175, 177,
179, 181, 200, 202, 215, 216, 220, 229,
237, 242, 295, 299, 302, 306, 311, 312
computer algebra system, see also Maple,
Mathematica, experimental mathematics,
198, 290, 307, 319
conditional convergence, xi, 10, 26, 39, 64, 118,
127, 137
conical lattice sums, 116
conjectures, 59, 65, 73, 77, 163, 194, 217, 219,
243, 314
contour integration, 63, 114, 146, 147, 217
Coulomb, Charles-Augustin de, 14, 30, 40, 215,
217, 229, 241
Crandall, R. E., xiii, 116, 118, 253
critical line, 144, 148, 150, 152, 153, 179
crystal structures, see also BCC, FCC, SC, etc.
CaF2 , 9, 40, 235, 236
CsCl, 29, 40, 117
NaCl, xi, xvii, 12, 15, 16, 40, 72, 88, 93, 104,
115, 117, 213, 238, 240, 247, 260
ZnS, 16, 117
cylinder, 126–128
de Wette, F. W., 26, 52, 64
Debye, Peter, 117
Dedekind, Richard
Dedekind eta function, 78, 204, 205, 217
Delves, R. T., 306, 309–311
diamond lattice, 234, 313
differential equation, 303, 307, 312, 319
differentiation under the integral sign, 143
dipole sums, 23, 26, 52, 126, 127, 231
Dirac, Paul
Dirac delta function, 114
Dirichlet, Johann, 55
365
Dirichlet L-series, 54, 56–58, 78, 117, 148,
150, 157, 166, 168, 178, 194, 195, 206,
213, 215, 220, 242, 250, 260, 290
table for 2D solutions, 60
table for 3D sums in terms of a single
L-series, 213
Dirichlet beta function, 28, 33, 39, 43, 47, 52,
66, 114, 115, 159, 183
Dirichlet character, see Dirichlet L-series
character table, 330
primitive character, 158, 187, 188
Dirichlet eta function, 33, 43, 116, 257
displaced lattice sum, 146, 164, 178
distributive lattice sum, 140, 142
divergence, 20, 39, 88, 94, 95, 100, 106, 109,
117, 148, 178, 227, 229
Domb, C., 297, 303
Eisenstein, G.
Eisenstein series, 178, 179, 197
electron sums, 226
ellipse, ellipsoid, 17, 250, 251, 259, 266
elliptic curve, xi, xiv, 217, 218
elliptic integral, 52, 66–70, 75, 77, 122, 171,
180, 218, 241, 295, 303, 304, 316, 317
K[N], 73
list of values for N from 1 to 100, 331
Emersleben, O., 15, 16, 27, 42, 54, 94
Epstein, P., 2–4, 10, 11, 17, 27
Epstein zeta function, 2, 15
Euler, Leonard, xiv, 204, 207, 211, 216, 217,
297
Euler totient function, 157
Euler–Mascheroni constant, 28, 52, 195
Evjen, H., 20, 88
Ewald, P., 9–11, 14–16, 20, 21, 23, 25, 30, 42,
44, 65, 88, 235
Ewald method, 9, 22, 233
experimental mathematics, xviii, 59, 76, 78,
118, 150, 163, 168, 175, 179, 194, 197,
219, 220, 243, 314, 319
face-centred cubic (FCC) lattice, 19, 21, 40, 70,
74, 228, 234, 295, 308, 312, 315
Faraday, Michael, 125
Ferguson, H., xi
Fermat, Pierre de, 200, 217
Feynman, Richard, 118, 314
five-dimensional sums, 220
fluorite, see crystal structures, CaF2
Foldy, L. L., 234, 235
Forrester, P. J., 245
four-dimensional sums, 34, 120, 195, 216, 223,
237
Fourier, Joseph, 14, 30
Fourier series, 78, 134, 135
Fourier transform, 2, 25, 311, 316
fast, 118
gamma function, 52, 66, 68, 74, 75, 78, 102,
117, 128, 180, 241, 290, 295, 296, 304
366
Index
incomplete gamma function, 22, 29
Gauss, Carl, 68, 150, 179, 208, 307, 320
Gauss circle problem, 90, 113, 261
Gaussian curvature, xi, xiii
Gaussian quadrature, 181
generating function, 67, 181, 319
Glasser, M. L., xviii, 30, 33, 47, 50, 59, 65–67,
72, 77, 131, 134, 258, 303, 304
Green, George
Green’s functions, 53, 70, 73, 130, 131, 133,
135, 137, 141, 164, 298, 310, 315
Guttmann, A. J., 74, 312
Hankel, Hermann, 43, 140, 141
Hardy, G. H., 28, 90, 113, 116, 146, 183, 250,
295
Helmholtz equation, 140
hexagonal close-packed (HCP) lattice, 233, 234
hexagonal lattice, 56, 72, 109, 229, 232, 233,
259
Hobson’s integral, 27, 132, 149, 221, 222
Hund, F., 15, 16, 19, 21, 27, 34–36, 39, 42, 233
Hund potentials, table, 37
Hurwitz, Adolf, 4
Hurwitz zeta function, 159, 162, 176, 290
hyperbolic function, 29, 44, 46, 47, 49, 63, 64,
128, 217, 218, 223, 232, 240
hypercube, 230, 231, 260, 313, 315
hypergeometric function, 68, 70, 73, 77, 136,
178, 216, 218, 220, 296, 303, 316, 319
Appell hypergeometric function, 301
transformation, 200, 306, 307, 311
inequality, 91, 96
triangle inequality, 112
integer relation program, see PSLQ
integration by parts, 18, 77, 272, 291, 316
invariant cubic lattice complexes (ICLCs), 32,
36, 39, 42
iPhone, 118
Iwata, G., 299, 301–303, 308, 310, 313
Jacobi, Carl, 3, 22, 27, 31, 32, 56, 114, 130, 132,
134, 158, 160, 171, 187, 203–205, 209,
210, 212, 216, 217, 296, 311
Jacobi symbol, see Kronecker symbol
Jacobi theta function, see theta function
Jacobi’s products Q i , 31, 204
triple product, 209
jellium, 137, 226
Joyce, G. S., 67, 68, 70, 73, 299, 303, 305, 310,
313
Kepler, Johannes, xiv
Klein, Felix, 77
Kronecker, Leopold, 4, 66, 138, 181
Kronecker’s Grenzformel, 66
Kronecker symbol, 56, 76, 158, 187, 197, 257
Lambert, Johann Heinrich
Lambert series, 160, 186, 189, 192, 197, 200,
209, 210
Landé, Alfred, xvii, 9
Landau, E., 17, 18, 251, 252, 267, 269
Laplace, Pierre-Simon
Laplace equation, 2, 6, 7, 126, 133, 135
Laplace transform, 26, 113, 216
Laplacian, 131, 298
lattice constants, xvii, 20, 72
lattice sum
definition of, 2, 248
table of exponential and hyperbolic forms, 49
table of numerical values, 51
table of sums, involving indefinite quadratic
forms, 176
Legendre, Adrien-Marie, 56, 190, 304
Legendre symbol, see Kronecker symbol
Liouville, Joseph, 153, 223
Lord Rayleigh, 125–127, 130, 134, 144
Lorentz H. A., 125
Lorenz, L., 20, 26, 28, 33, 113, 125, 146, 175,
183
Macdonald functions, see modified Bessel
functions
Madelung, E., xvii, 4, 7, 8, 12, 20, 28–30, 43,
117, 118, 238
Madelung constant, xi, xvii, 12, 15, 16, 36,
72, 87, 88, 93–95, 100, 101, 104, 109,
111, 114, 116, 118, 184, 217, 237, 247,
259, 260, 290
table of Madelung constants for ionic
crystals, 41
magnetism, 9, 68, 295
Mahler, K.
Mahler measure, 73, 78, 219, 314
Maple, 125, 130, 179, 199, 319
Maradudin, A. A., 29, 226, 229, 295
Mathematica, 76, 125, 130, 169
McPhedran, R. C., 138, 140, 148, 164, 168, 170
mean value theorem, 91
Mellin, H.
Mellin transform, 30, 32, 36, 65, 78, 101, 102,
110, 132, 149, 151, 152, 160, 164, 184,
186, 189, 190, 192–194, 206, 207,
209–211, 216, 221, 223, 239, 285, 290
of θ -function (MTθ F), 30
table, 209
method of expanding geometric shapes, xiii, 89,
94, 95, 106, 109, 249, 259, 261, 263
modular equation, 186, 193–195, 197, 199, 211,
305, 325–329
modular equation order, 3, 110, 188,
190–193, 211
tables for order 3, 325–327
tables for order 5, 328
tables for order 7, 329
modular form, 55, 74, 78, 197, 217, 219, 273
modular group, 197
monotonicity, 91, 93, 95, 96, 98, 268, 289
Index
Movchan, A. B., 138
MTθ F, 30
multipole, 16, 25, 26, 126–128
Naor potentials, table, 37
Newton, Isaac, 183
Newton-John, Olivia, xvii
Nicorovici, N. A., 138
Nijboer, B. R. A., 25, 26, 30, 52
number theory, xiv, 18, 54, 56, 89
numerical methods, xviii, 4, 16, 20, 29, 30, 39,
42, 44, 51, 54, 65, 72, 128, 142, 146,
148–151, 175, 178, 179, 181, 219, 233,
241, 247, 297, 313, 315
optical properties, 26, 125
orthogonal polynomials, 181
orthonormalization, 183
orthorhombic lattice, 50, 310
Parseval’s theorem, 25
partial fractions, 178, 290
Pathria, R. K., 30, 43, 47, 50, 52
perovskite, 72, 235
phase-modulated sums, 131, 164
Pochhammer, L. A.
Pochhammer symbol, 73, 139, 178, 291, 311
Poisson, Siméon Denis, 182
Poisson equation, 6, 8, 237, 315
Poisson summation formula or transform, 7,
29, 30, 43, 44, 45, 50, 132, 140, 142,
149, 164, 190–192, 205, 207, 211, 215,
221, 227
table, 209
pole, simple, 4, 39, 45, 63, 142, 145, 147
power series, 128, 176, 181, 302
prime number, 54, 56, 118, 153, 157, 160, 161,
187, 218
probability, 151, 296
PSLQ algorithm, xi, 179, 199, 243, 320
quadratic form, 2, 18, 55, 56, 58, 113, 159, 166,
175, 177, 186, 193, 216, 236, 248–250,
252, 254, 256, 258, 264, 283
discriminant, 55, 56, 58, 59, 76, 116, 159,
175, 177, 258
genus, 58, 59, 177, 194, 258
indefinite, 175
quadratic residue, see Kronecker symbol
Ramanujan, S., 73, 78, 116, 171, 178, 186, 189,
193, 199, 200, 208, 211, 212, 304, 349
random walk, 67, 68, 78, 152, 294, 296, 302,
306
reciprocal lattice, 3, 9, 10, 13, 23, 42, 131, 134,
135, 139–141, 144
Brillouin zone, 67, 146–148
rectangular lattice, 7, 131, 151
recurrence, 23, 129, 137–139, 151, 180, 182,
291, 303, 307, 319
recursion, see recurrence
367
restricted sums, 112
rhombohedral lattice, 29
Riemann, Bernhard, xiii, 3, 9
generalized Riemann hypothesis, 148
Riemann hypothesis, 116, 147, 148, 151, 153,
179
Riemann zeta function, 2, 45, 54, 110, 125,
152, 159, 180, 183, 188, 220, 223, 228,
242, 290
Robertson, M. M., 34, 54, 59, 157, 158, 161
rock salt, xvii, 4, 8, 28, 217, see also crystal
structures
Rogers, Mathew, 74, 78, 178, 218, 219, 315
Sakamoto, Y., 39, 42
Selberg, A., 66, 75
series, slowly convergent, 7, 10, 25, 50, 118
sign transform, 186, 190, 192, 204, 207, 211,
212
simple cubic (SC) lattice, 14, 15, 21, 22, 40, 67,
70, 74, 126, 226, 228, 234, 235, 237, 295,
302, 306, 310, 313, 315
singular value, 68, 70, 75, 304, 317, 318
k210 , 116
table of singular values of K [N ], 331
spanning tree, 72, 314
spinel, 72, 235
square lattice, 72, 114, 127, 129, 134, 135, 138,
140, 144–146, 164, 232, 316
Stirling, James
Stirling numbers, 291
symmetry, mathematical, 5, 15, 21, 26, 28
tables
L-series solutions of 2D sums, 60
2D Bessel sums, 44
3D sums in terms of a single Dirichlet
L-series, 213
character table, 330
exponential and hyperbolic forms for lattice
sums, 49
expressions in Eulerian and Jacobian forms,
208, 210
lattice sums involving indefinite quadratic
forms, 176
Madelung constants for ionic crystals, 41
modular equations of order 3, 325–327
modular equations of order 5, 328
modular equations of order 7, 329
Naor and Hund potentials, 37
numerical values of lattice sums, 51
Poisson and Mellin transforms, 209
singular values of K [N ], 331
Taylor series, 130, 131, 145, 151, 182, 198, 199,
295
term-by-term integration, 241, 262, 306, 308
test
alternating series, 289
comparison, 101
Weierstrass M-test, 108, 263, 265
368
Index
tetragonal lattice, 50, 306
theta functions, 22, 30, 34, 87, 110, 113, 114,
130, 132, 134, 147, 160, 165, 178, 184,
186, 200, 203, 216, 241
conical, 116
multiplicative relations, 32
table of Eulerian and Jacobian forms, 208,
210
θ1 , 66, 134
θ1 , 31, 39, 66, 134, 212
θ2 , 34, 36, 39, 45, 46, 78, 103, 104, 189, 211,
223
θ3 , 32, 34, 36, 39, 45, 46, 78, 189, 193, 195,
210, 211
θ3 functional equation, see Poisson
summation formula
θ4 , 32, 34, 39, 45, 46, 78, 101, 112, 189, 192,
211
θ5 , 31, 34, 45, 186, 189, 203
θ6 , 31
three-dimensional sums, 34, 39, 52, 94, 184,
202, 215, 221, 232, 240, 256, 292
Tosi, M. P., 1, 16, 21, 29, 40
tree, see spanning tree
triangular lattice, see hexagonal lattice
two-dimensional lattice sum, 55, 57, 59, 60, 89,
105, 109, 160, 177, 178, 184, 193, 256, 292
Vandermonde, A., 183
Wan, J. G., 78, 184
Watson, G. N., 31, 43, 53, 67, 270, 294, 298,
301, 302, 304
Watson integrals, 53, 67–70, 73, 294
wave, 140–142
Weber’s integral, 143
Weierstrass, Karl, 130
Weierstrass elliptic function, 54, 130, 180
Weierstrass M-test, 108, 263, 265
Wigner, E. P., 150, 226, 236
Wiles, Andrew, 217, 219
WIs, see Watson integrals
Wu, F. Y., 72
Zeilberger, D., 319
Zucker, I. J., xviii, 32, 33, 45, 47, 50, 52, 54, 59,
64–68, 75, 78, 147, 157, 158, 161, 170,
194, 212, 221, 229, 235, 258, 288, 304, 313
Zudilin, W., 218, 219