Quantum Physics and Number Theory
connected by
the Riemann Zeta Function
Dissertation
zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakultät für Naturwissenschaften
der Universität Ulm
vorgelegt von
Cornelia Feiler
aus Ulm
Institut für Quantenphysik
Ulm, 2013
Amtierender Dekan:
Prof. Dr. Joachim Ankerhold
Erstgutachter:
Prof. Dr. Wolfgang P. Schleich
Zweitgutachter:
Prof. Dr. Peter Reineker
Tag der Prüfung:
18.02.2014
Contents
List of figures
vi
List of tables
vi
List of symbols
vii
Introduction
1. The
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1
Riemann zeta function: mathematical representations
Dirichlet series . . . . . . . . . . . . . . . . . . . . . . .
Alternating sum . . . . . . . . . . . . . . . . . . . . . .
Analytical continuation . . . . . . . . . . . . . . . . . .
Symmetric representation . . . . . . . . . . . . . . . . .
Riemann-Siegel formula . . . . . . . . . . . . . . . . . .
Extended Berry-Keating formula . . . . . . . . . . . . .
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3
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2. The Riemann zeta function: topology
2.1. Continuous Newton method: essentials . .
2.2. Newton flow of the Riemann zeta function
2.2.1. Asymptotic for σ → +∞ . . . . . .
2.2.2. Asymptotic for σ → −∞ . . . . . .
2.2.3. Non-trivial separatrices . . . . . .
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9
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3. Combined quantum systems
3.1. Time evolution . . . . . . . . . . .
3.2. Joint measurement . . . . . . . . .
3.3. Phase space representations . . . .
3.3.1. Wigner and Moyal function
3.3.2. Wigner and Moyal matrix .
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15
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4. Riemann states
4.1. Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Wigner functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Moyal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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25
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5. Alternating sum and truncated Riemann states
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5.1. Exact solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.1. Construction à la Riemann states . . . . . . . . . . . . . . . . . . . 29
i
Contents
5.1.2. Entanglement . . . . . . . .
5.2. Truncated Riemann states . . . . .
5.3. Truncated alternating sum . . . . .
5.3.1. Reference state and overlap
5.3.2. Wigner representation . . .
5.3.3. Moyal representation . . . .
5.4. Summary . . . . . . . . . . . . . .
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6. Riemann-Siegel states
6.1. Definition of the states . . . . . . . . . . . .
6.1.1. Riemann-Siegel state . . . . . . . . .
6.1.2. Riemann-Siegel reference state . . .
6.2. Entanglement . . . . . . . . . . . . . . . . .
6.2.1. Riemann-Siegel state . . . . . . . . .
6.2.2. Riemann-Siegel reference state . . .
6.3. Wigner matrix of the Riemann-Siegel state
6.4. Wigner matrix of the reference state . . . .
6.5. Overlap in Wigner representation . . . . . .
6.6. Overlap in Moyal representation . . . . . .
6.7. Summary . . . . . . . . . . . . . . . . . . .
7. Berry-Keating reference states
7.1. Definition . . . . . . . . . . . . .
7.2. Wigner function of Berry state .
7.3. Wigner matrix . . . . . . . . . .
7.4. Overlap in Wigner representation
7.5. Overlap in Moyal representation
7.6. Summary . . . . . . . . . . . . .
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8. Jaynes-Cummings-Paul approach to the Riemann
8.1. Jaynes-Cummings-Paul model . . . . . . . . .
8.2. Riemann Hamiltonian . . . . . . . . . . . . .
8.2.1. Exact solution . . . . . . . . . . . . .
8.2.2. Approximate solution . . . . . . . . .
8.3. Summary . . . . . . . . . . . . . . . . . . . .
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Hamiltonian
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30
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Summary
79
A. Special functions
A.1. The gamma function Γ . . . . . . . . . . .
A.1.1. Definition and functional equation
A.1.2. Asymptotics . . . . . . . . . . . .
A.1.3. Formulae . . . . . . . . . . . . . .
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84
ii
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Contents
A.2. The function χ(s) . . . . .
A.2.1. Definitions . . . .
A.2.2. On the critical line
A.2.3. Asymptotics in the
A.3. The function ϑ(s) . . . . .
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critical strip
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B. Approximations in the critical strip
B.1. Riemann-Siegel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2. Berry-Keating formula on the critical line . . . . . . . . . . . . . . . . . .
B.3. Berry-Keating formula in the critical strip . . . . . . . . . . . . . . . . . .
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92
C. Continuous Newton method
C.1. Newton flow . . . . . . . . . .
C.1.1. Sink and source . . . .
C.1.2. Origin of separatrices
C.2. Further examples . . . . . . .
C.2.1. The function χ . . . .
C.2.2. The function Fe . . .
C.3. Technical details . . . . . . .
C.4. Newton flow of ζ . . . . . . .
C.5. Zeros of the derivative ζ 0 . . .
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95
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102
102
D. Phase space representations
D.1. Properties of the Wigner function . .
D.2. Moyal functions . . . . . . . . . . . .
D.3. Wigner matrix . . . . . . . . . . . .
D.4. Fock states . . . . . . . . . . . . . .
D.4.1. Wigner and Moyal functions
D.4.2. Sum over Wigner functions .
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113
E. Entanglement
E.1. Quick check . . . . . . . . . . .
E.2. Schmidt decomposition . . . . .
E.3. Wigner representation . . . . .
E.4. Riemann-Siegel reference state
E.5. Berry-Keating reference state .
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115
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119
F. Normalization and proportionality factors
F.1. Truncated Riemann state . . . . . . .
F.2. The function E(n, τ ) . . . . . . . . . .
F.3. Berry state . . . . . . . . . . . . . . .
F.4. Truncated alternating sum . . . . . . .
F.5. Riemann-Siegel reference state . . . .
F.6. Berry-Keating reference state . . . . .
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121
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iii
Contents
G. Effective Hamiltonian
129
G.1. Time-dependent Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 129
G.2. Interaction picture and effective Hamiltonian . . . . . . . . . . . . . . . . 130
H. Matrix elements
133
†
H.1. Calculation with â and â . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
H.2. General expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
H.3. Explicit expression up to fourth order . . . . . . . . . . . . . . . . . . . . 136
Bibliography
137
Zusammenfassung
143
Publikationen
147
Poster und Vorträge
149
Curiculum vitae
151
Danksagung
153
iv
List of figures
1.1. Absolute value of Riemann zeta function ζ and sketch of its structure . .
1.2. Symmetric representation Z and its approximations on the critical line . .
1.3. Z and its approximations for different σ and large τ . . . . . . . . . . . .
4
7
8
2.1. Newton flow of the Riemann zeta function . . . . . . . . . . . . . . . . . . 11
2.2. Separatrices of ζ and χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1. Wigner function of the initial Riemann state W|σ,0i for different σ . . . . . 23
4.2. Time evolution of W|2,τ i and overlap |h2, 0|2, τ i|2 . . . . . . . . . . . . . . 24
4.3. Time evolution of the Moyal function W|2,τ ih2,0| . . . . . . . . . . . . . . . 26
5.1. Wigner function of the initial truncated Riemann state W|1/2,0iν for different ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Time evolution of W|2/3,τ i100 and overlap |100 hφ|2/3, τ i100 |2 . . . . . . .
5.3. Truncated alternating sum ζN for different σ and N for small τ . . . . .
5.4. Truncated alternating sum ζN for different σ and N for large τ . . . . .
5.5. Time evolution of the Moyal function W|2/3,τ i100 hφ| . . . . . . . . . . . .
6.1. Time evolution of Ŵ|ΨRSi of the Riemann-Siegel state
6.2.
6.3.
6.4.
6.5.
6.6.
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Wigner matrix elements of Ŵ|ΦRSi of the Riemann-Siegel reference state
Wigner representation KRS of overlap |hΦRS|ΨRSi|2 . . . . . . . . . . . . .
Time evolution of the Moyal functions W|1/3,τ i10 h1/3,0| and W|1/3,−τ i10 h1,0|
Moyal representation K RS of overlap hΦRS|ΨRSi . . . . . . . . . . . . . . .
Riemann-Siegel formula ζRS for σ = 1/3 and σ = 1/2 . . . . . . . . . . .
Argand diagram of ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
7.1. Difference W|φB(1/2,τ )in0 − W|1/2,0in0 between Wigner functions of Berry
state |φB(σ, τ )in0 and initial truncated Riemann state |σ, 0in0 . . . . . . .
7.2. Wigner matrix elements of Ŵ|ΦBKi of the Berry-Keating reference state
and difference to fig. 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3. Wigner representation KBK of overlap |hΦBK|ΨRSi|2 and difference to fig. 6.3
7.4. Moyal functions W|1/3,τ i10 hφB(1/3,τ )| and W|1/3,−τ i10 hφ (1,τ )| and difference
B
to fig. 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5. Moyal representation K BK of overlap hΦBK|ΨRSi and difference to fig. 6.5 . .
47
49
52
53
55
55
59
61
63
65
66
8.1. Two-level atom in cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.2. ζ displayed by approximated Hamiltonian . . . . . . . . . . . . . . . . . . 76
v
C.1.
C.2.
C.3.
C.4.
Newton
Newton
Newton
Newton
flow
flow
flow
flow
of
of
of
of
simple zero, simple pole and Möbius transformation .
polynomials with two and three zeros . . . . . . . . .
χ and Fe . . . . . . . . . . . . . . . . . . . . . . . . .
ζ for imaginary parts up to τ = 300 . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
96
98
100
103
D.1. Wigner functions W|ni of Fock states . . . . . . . . . . . . . . . . . . . . . 111
D.2. Real part of Moyal function W|nihm| of Fock states . . . . . . . . . . . . . 112
D.3. Sum SN of Wigner functions W|ni . . . . . . . . . . . . . . . . . . . . . . 113
F.1.
F.2.
F.3.
F.4.
F.5.
Square of normalization Nν of truncated Riemann states . . .
Behavior of E(n, τ ) in dependence on n for fixed τ . . . . . .
en of Berry states . . . . . . . . . .
Square of normalization N
B
(N )
Proportionality factor Ma of the truncated alternating sum
Factors in Ŵ|ΦRSi of the Riemann-Siegel reference state . . . .
. .
. .
. .
ζN
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
121
122
123
124
126
F.6. Factors in Ŵ|ΦBKi of the Berry-Keating reference state . . . . . . . . . . . 127
List of tables
2 . . . . . . . . . . . . . . 50
6.1. Analysis of fig. 6.3 – Wigner representation of ζRS
6.2. Analysis of fig. 6.5 – Moyal representation of ζRS . . . . . . . . . . . . . . 54
2 . . . . . . . . . . . . . . 62
7.1. Analysis of fig. 7.3– Wigner representation of ζBK
7.2. Analysis of fig. 7.5 – Moyal representation of ζBK . . . . . . . . . . . . . . 68
C.1. Trivial zeros of the derivatives ζ 0 and χ0 . . . . . . . . . . . . . . . . . . . 104
C.2. Non-trivial zeros of ζ 0 with τ < 100 and arg ζ at these points . . . . . . . 104
C.3. Non-trivial zeros of ζ 0 with imaginary part 100 < τ < 300 . . . . . . . . . 105
H.1. Matrix elements hn ± j| (â + ↠)µ |ni for n = 0, . . . , 4. . . . . . . . . . . . . 134
vi
List of symbols
|ψi
|Ψi
|σ, τ i
|σ, τ iν
|φBi
|ΨRSi
|φiN
|ΦRSi
|ΦBKi
state of one system
state of two systems
Riemann state
truncated Riemann state
Berry state
Riemann-Siegel state
reference state for truncated alternating sum
Riemann-Siegel reference state
Berry-Keating reference state
χ
C
C
function, factor in functional equation of ζ
overlap of time-evolved state and reference state
set of complex numbers
Erfc
complementary error function
γ
γB
Γ
factor in Riemann-Siegel reference state
factor in Berry-Keating reference state
Gamma function
~
Ĥ0
ĤaJC
Ĥa
Ĥfield
I
Ĥint
Ĥint
ĤJC
ĤR
Planck constant
free Hamiltonian
anti-Jaynes-Cummings Hamiltonian
free Hamiltonian of two level atom
free Hamiltonian of cavity field
interaction Hamiltonian in interaction picture
interaction Hamiltonian in Schrödinger picture
Jaynes-Cummings Hamiltonian
Riemann Hamiltonian
K
KRS
KBK
K
K RS
K BK
kernel
kernel
kernel
kernel
kernel
kernel
Ma
(ν,N )
Ma
MRS
proportionality factor for alternating sum
proportionality factor for truncated alternating sum
proportionality factor for Riemann-Siegel formula
of
of
of
of
of
of
Wigner representation
Wigner representation for Riemann-Siegel formula
Wigner representation for Berry-Keating formula
Moyal representation
Moyal representation for Riemann-Siegel formula
Moyal representation for Berry-Keating formula
vii
List of symbols
MBK
proportionality factor for Berry-Keating formula
n̂
n0
nB
N
Nν
Nn0
en
N
B
~
N
N
N\{0}
photon number operator
summation limit in Riemann-Siegel formula
summation limit in Berry-Keating formula
norm of Riemann state |σ, τ i
norm of truncated Riemann state |σ, τ iν
norm of Riemann-Siege reference state
norm of Berry-Keating reference state
vector of the Newton flow
set of natural numbers including 0
set of natural numbers without 0
ω
Rabi frequency
σ̂z
σ
s
Pauli spin matrix
real part of complex variable s
complex variable
τ
ϑ
imaginary part of complex variable s
function, derived from χ
W|ψi
W|ψihφ|
Ŵ|Ψi
Wigner function of state |ψi
Moyal function of |ψihφ|
Wigner matrix of state |Ψi
ζ
ζBK
ζN
ζRS
Z
ZBK
ZRS
Z
Riemann zeta function
Berry-Keating formula
truncated alternating sum of ζ
Riemann-Siegel formula
symmetric representation of ζ
symmetric representation of ζBK
symmetric representation of ζRS
set of integers
viii
Introduction
Number theory is not pure Mathematics.
It is the Physics of the world of Numbers.
Alf van der Poorten1
Number theory concentrates on the fundamentals of mathematics: integers or, more
precisely, primes. Questions concerning the properties of primes influence all branches of
mathematics and therefore also physics. Even our lives are affected by prime numbers,
for example when we think of the security of codes, in bank transfer or communication,
which are based on prime factorization.
On the other hand, the distribution of primes [2] is intimately connected to the distribution of the non-trivial zeros of the Riemann zeta function ζ [3–5]. Since the location
of these zeros, which is object to the Riemann hypothesis, is by now unconfirmed, many
efforts are directed to the investigation of ζ, not only with mathematical but also with
physical methods [6]. For example, the non-trivial zeros can be considered as eigenvalues
of an Hermitian operator [7, 8], while the phase of ζ on the critical line is closely related
to the scattering on an inverted harmonic oscillator [9]. Moreover, the Riemann zeta
function emerges as a partition function of a quantum dynamical system [10], from wave
packed dynamics in a potential with logarithmic energy spectrum2 [13], or a quantum
Mellin transform [14].
In this thesis, we consider several representations of the Riemann zeta function to gain
insight into its characteristics. Therefore, we shortly recall the properties and different
representations derived from the definition in chapter 1, and introduce the continuous
Newton method to visualize the topology of ζ by lines of constant phase in chapter 2.
The corresponding Newton flow particularly illuminates the location of special points:
the single pole as well as the zeros of ζ and its derivative ζ 0 .
Chapter 3 proposes a physical approach to the Riemann zeta function (see also [15]).
It is motivated by the similarity of the phases in the Dirichlet series of ζ to the time
evolution of an entangled state in cavity quantum electrodynamics, described by the
Jaynes-Cummings-Paul model [16,17]. Yet, in our system, we have to choose the dependence of the Hamiltonian on the number operator n̂ logarithmically instead of linear.
The resulting states reproduce the different representations of the Riemann zeta function when we take the overlap of the time-evolved state with an appropriately chosen
reference state, thus making the values of ζ accessible to measurement. Moreover, we
remind of the Wigner and Moyal representations [18–21] which are used to describe the
behavior of the states and the overlap in phase space.
1
2
quoted by M. Planat in [1]
Recently was shown that two [11] or more [12] bosons in this potential allow us to factor numbers.
1
Introduction
The following chapters are dedicated to the description of the different zeta states and
their phase space representations, continuing the work of the diploma thesis [22]. In
chapter 4, we define the Riemann state |σ, τ i of the cavity whose overlap with its initial
state produces the Dirichlet representation of ζ(σ + iτ ). In this case, entanglement is
not necessary. However, the states are restricted to the region of complex space where
the real part σ of the argument s is larger than one, in agreement with the limitation of
the Dirichlet sum.
Mathematically, we can overcome the restriction σ > 1 with the alternating sum
representation of ζ which is convergent for all positive values of σ. But the states
producing the exact sum underlie the same limitation as the Riemann states, as we show
in the first section of chapter 5. Nevertheless, we can use the alternating representation
when we truncate the summation limit. In the physical picture, this yields the truncated
Riemann states |σ, τ iν which are normalizable as long as the truncation parameter ν is
finite. Even so, the normalization constant makes the probability amplitudes in |σ, τ iν
small for large ν. Consequently, an improvement of the approximation by increasing the
summation limit results in very small values for the overlap of the states.
The approximations by the Riemann-Siegel and Berry-Keating formula finally allow
the definition of states which can reach into the critical strip, as shown in the chapters 6
and 7. There, we take advantage of the similar structure of the formulae and use in
both cases the Riemann-Siegel state |ΨRS(t)i as a time-evolved state. It consists of two
counter-rotating truncated Riemann states coupled to the excited and ground state of a
two-level atom. This structure is reminiscent of the Schrödinger cat state [23,24] realized
experimentally in high-Q cavities [25,26] or with ions stored in a trap [27]. In contrast to
the Schrödinger cat state, the phases of the Riemann-Siegel state depend logarithmically
on the photon number n, due to the Riemann Hamiltonian ĤR .
The differences of the formulae are reflected in the properties of the corresponding
reference states. Although these differences are small, they result in the fact that the
Berry-Keating reference state |ΦBKi is always entangled whereas the Riemann-Siegel
reference state |ΦRSi becomes separable on the critical line σ = 1/2.
The last chapter gives a short review of the Jaynes-Cummings-Paul model and suggests how we can modify this model to produce the Riemann zeta function with an
approximated Hamiltonian.
Finally, the appendices contain supplements to the chapters: Appendices A and B
summarize the properties of the functions involved in the different representations of ζ,
which are presented in chapter 1. The characteristics of the Newton flow are derived
and illustrated by examples in appendix C. In appendix D, we recall the different
phase space representations obtained from the familiar Wigner function and describe
the phase space functions of the Fock states in more detail, since all zeta states are
defined in this basis. Appendix E reminds of different tests for entanglement and
contains proofs concerning the entanglement of the Riemann-Siegel and Berry-Keating
reference state. The following appendix lists the properties of the normalization and
proportionality factors occurring in the different states, while the last two appendices
accomplish the analysis of chapter 8.
2
1. The Riemann zeta function:
mathematical representations
In this chapter, we give a short overview of the different representations of the Riemann
zeta function ζ and their properties. Although not every representation is suited for our
purpose to realize ζ as the outcome of a joint measurement of two quantum states, we
nevertheless recall them for the sake of completeness.
1.1. Dirichlet series
The definition of the Riemann zeta function [4] by the Dirichlet series
∞
X
1
ζ(s) ≡
ns
(1.1)
n=1
for a complex variable s was already known to Leonhard Euler in 1737. He proved the
connection
∞
X
Y
1
1
=
(1.2)
s
n
1 − p−s
n=1
p prime
between ζ and the infinite product over all prime numbers p. However, both representations suffer from the restriction to the area σ ≡ Re s > 1. Needless to say, this restriction
will also affect the quantum states we derive from the Dirichlet series in chapter 4.
In this region, the zeta function has no zeros. This becomes clear when we recall
that the product on the right hand side of eq. (1.2) consists only of non-zero factors.
Moreover, for s = 1 the zeta functions has one simple pole since the Dirichlet sum merges
into the harmonic series.
1.2. Alternating sum
A simple way to expand the domain of definition is subtracting from the Dirichlet series
twice the sum over all even terms
∞
∞
∞
X
X
X
1
1
(−1)n
−
2
=
−
ns
(2n)s
ns
n=1
n=1
n=1
which yields an alternating sum convergent for σ > 0. The left hand side of this equation
is equal to (1 − 21−s ) ζ(s). Therefore, ζ can be represented by the alternating sum
ζ(s) ≡
1
21−s − 1
∞
X
(−1)n
n=1
ns
(1.3)
3
1. The Riemann zeta function: mathematical representations
Figure 1.1.: The left picture shows |ζ(s)| with s = σ + iτ and the right one a sketch of its
significant points. The only pole at s = 1, marked in the sketch by a dot, is cut on the left
picture to show the structure of the zeta function around this point more clearly. The blue line
denotes the real axis on which the trivial zeros are located at s = −2k, k ∈ N\{0}, whereas the
red line indicates the critical line σ = 1/2, conjectured location of the non-trivial zeros [3]. All
zeros are marked in the sketch by crosses and the critical strip 0 < σ < 1 is indicated by the
shaded area.
in the whole right of the complex plane s. We will see in chapter 5 that this representation
is not as powerful in our physical picture as it is in mathematics, since the constructed
states are only normalizable for σ > 1. For this reason, we need other representations
to reach beyond the wall at σ = 1.
1.3. Analytical continuation
In 1859, Bernhard Riemann published his seminal paper “Ueber die Anzahl der
Primzahlen unter einer gegebenen Grösse” [3]. Although he used the zeta function
merely as a tool to determine the number of primes beneath a given value, he was the
first who gave an analytical continuation of ζ valid in the whole complex plane. In this
paper, he derived the integral representation


Z∞


s/2
s−2
π
1
1
− s+1
2 + x 2
ζ(s) ≡
−
+
dx
x
w(x)
(1.4)

Γ( 2s )  s − 1 s
1
with
w(x) ≡
from which the functional equation
∞
X
2x
e −πn
,
n=1
ζ(s) = χ(s) ζ(1 − s)
4
(1.5)
1.4. Symmetric representation
immediately follows. In literature [4,28,29], the function χ(s) is defined in different ways,
whose equivalence we show in appendix A.2. In what follows, we use the representation
[28]
1−s
s− 12 Γ
2
χ(s) ≡ π
,
(1.6)
Γ 2s
where Γ(s) denotes the Gamma function.
Recalling that the zeta function has no zeros in the region σ > 1, the functional
equation (1.5) reveals that the only zeros for σ < 0 are given by χ(s) or precisely by
the poles of Γ(s/2) in the denominator of definition (1.6). The poles are located at
s = 0, −2, −4, . . . and describe the so-called trivial zeros of ζ at s = −2k, k ∈ N\{0}.
At s = 0, we find a removable singularity, since the pole of Γ is compensated by the pole
s−1 in eq. (1.4).
Hence, all other zeros have to lie in the critical strip 0 < σ < 1, indicated in the right
picture of fig. 1.1 by the shaded area. Due to the functional equation (1.5), they must
be located symmetrically to the axis σ = 1/2, the so-called critical line. Moreover, they
are symmetric to t = 0. This follows from the fact that if z0 ∈ C is a zero its complex
conjugate z̄0 is also one.
The famous Riemann hypothesis [3] states that
all non-trivial zeros of ζ have real part 1/2.
However, a rigorous proof of this claim still does not exist, but the hypothesis was already
confirmed for the first 1.5 billion zeros [30].
1.4. Symmetric representation
Another useful expression provides the symmetric function
Z(s) ≡ χ−1/2 (s) ζ(s) ,
(1.7)
which with the help of
χ(s) = χ−1 (1 − s)
transforms the functional equation (1.5) into
Z(s) = Z(1 − s) .
(1.8)
This expression reveals the convenient property that Z is real on the critical line. Moreover, the zeros of Z are given by the trivial and non-trivial zeros of ζ, but due to the
symmetry, there are also zeros at s = 2n + 1 and an additional pole at s = 0.
Needless to say, the functional equations (1.5) and (1.8) can only describe the zeta
function in the critical strip when we use a representation of ζ valid there, for example
the integral representation given by eq. (1.4). However, the complexity of this formula is
quite difficult to realize with our physical approach. Therefore, we will use in this thesis
the famous expressions of Riemann-Siegel [29] and Berry-Keating [31], introduced in the
following sections, to find states for an approximate description of ζ in this region.
5
1. The Riemann zeta function: mathematical representations
1.5. Riemann-Siegel formula
Bernhard Riemann mentioned in a letter to Karl T. W. Weierstraß in 1859 that he had
found a new expansion of the zeta function, but he had not yet simplified its derivation
enough to include the formula in his article on prime number theory. Fortunately, the
unpublished representation was found amongst his ‘Nachlass’ and rederived by Carl L.
Siegel [29]. Therefore, it was named Riemann-Siegel formula.
The main part of this representation, independently rediscovered by Godfrey H. Hardy
and John E. Littlewood [32] in 1920, is given by
ζ(s) ∼
= ζRS(s) ≡
n0
n0
X
X
1
1
+
χ(s)
,
s
1−s
n
n
n=1
(1.9)
n=1
where χ(s) is defined by eq. (1.6) and the summation limit
r τ
n0 ≡
2π
(1.10)
depends on the imaginary part τ of s. Here, the floor function (or Gauß bracket) [x]
indicates the largest integer smaller or equal to x. The correction term to the main part
ζRS is given in appendix B.
The Riemann-Siegel formula, eq. (1.9) provides a reliable approximation for large τ ,
especially in the critical strip 0 < σ < 1. However, it suffers from the fact that, by
construction, the function jumps at τ = 2π n20 .
Before we introduce the Berry-Keating formula, which avoids these discontinuities, we
use eq. (1.9) in eq. (1.7) to get
n0
X
2
1
√ cos ϑ(s) + i s −
ZRS(s) ≡
ln n
(1.11)
2
n
n=1
as Riemann-Siegel approximation of the symmetric representation. The function ϑ(s) is
defined by
e iϑ(s) ≡ χ−1/2 (s) .
Equation (1.11) clearly shows that Z is real on the critical line.
1.6. Extended Berry-Keating formula
Michael V. Berry and Jonathan P. Keating [31] derive a series of convergent sums as
an approximation of the symmetric representation Z, eq. (1.7), on the critical line,
which is, according to them, term by term more accurate than the Riemann-Siegel
formula. The proof that the main term of this series already contains the Riemann-Siegel
representation ZRS, eq. (1.11), and its first correction term is given in appendix B.2.
The Berry-Keating formula in the critical strip (see appendix B.3) reads
"
#
nX
B −1
χ−1/2 (s) E(n, τ ) χ1/2 (s) E (n, τ )
ZBK(s) =
+
(1.12)
(n + 1)s
(n + 1)1−s
n=0
6
1.6. Extended Berry-Keating formula
Z(1/2 + iτ )
4
2
20
30
40
50
60
τ
-2
-4
Z(1/2 + iτ )
Z(1/2 + iτ )
1.0
3
2
0.5
1
14
14.5
15
τ
0
56
57
58
59
60
τ
-1
-0.5
-2
Figure 1.2.: The upper plot shows Z(s), eq. (1.7), and ZRS(s), eq. (1.11), indicated by the red
and black dotted line, respectively, in comparison to ZBK(s), eq. (1.12), given by the blue dashed
line, for σ = 1/2 and nB = 4. The Berry-Keating representation ZBK excellently fits the exact
curve of Z. Only high resolution in the lower pictures reveals a tiny deviation for small values
of τ . In contrast, the approximation by the Riemann-Siegel formula ZRS(s) is worse and jumps
at τ = 2πn20 caused by the jumps of n0 , marked by the orange line.
7
1. The Riemann zeta function: mathematical representations
Figure 1.3.: For large values of τ , one can barely distinguish between the exact curve Z(s) and
ZBK(s) indicated by the red and blue dashed line, respectively. Even for σ 6= 1/2 ZBK is a better
approximation than ZRS(s) (black dotted line), which even misses two zeros for τ around 220
(left picture). Here, we have used nB = 8 while n0 = 5.
with τ ≡ Im s, nB ≥ n0 and the abbreviation
#
"r
τ
n+1
1
ln p τ
.
E(n, τ ) ≡ Erfc
2
1−i
2π
(1.13)
Here, Erfc denotes the complementary error function defined in eq. (B.8). In the original
definition, Berry and Keating use an infinite sum, but cutting it at nB is convenient and
has no influence on the results as convergence considerations prove [31]. Moreover,
the truncation parameter nB does not depend on τ . Thus, the discontinuities of the
Riemann-Siegel formula are eliminated.
Figures 1.2 and 1.3 confirm that ZBK is a more accurate approximation of Z compared
to the Riemann-Siegel formula ZRS, eq. (1.11). Even for small values of τ , the blue dashed
curve of ZBK and the red one of Z fit quite perfectly. Only high resolution in the bottom
pictures reveals small deviations. For large τ , one can barely distinguish between these
two lines, whereas the black dotted line of ZRS even misses two zeros in the left picture
of fig. 1.3. The orange line indicates the values of n0 and the jumps of ZRS at τ = 2π n20 .
Before we use the different representations of ζ to construct quantum states reproducing – with an appropriately chosen reference state – the values of the Riemann zeta
function by a joint measurement, we describe a method to find numerically the zeros
of a complex function. This method is applied to the Riemann zeta function to give a
flavor of its topology with respect to the phase of its values.
8
2. The Riemann zeta function: topology
This chapter introduces the continuous Newton method, which is designed to find numerically the zeros (or poles) of a complex function. We demonstrate its benefits by the
example of the Riemann zeta function. In contrast to the previous chapter, the topology
of ζ is not described by contour lines, that is lines with constant absolute value, but
with the help of the Newton flow consisting of lines with constant phase.
Already in 1933, Eugene Jahnke and Fritz Emde [33] depicted complex functions by
lines of constant phase. Later, Albert A. Utzinger, Andreas Speiser and Juan Arias-deReyna [34–36] concentrated on the description of real lines to get a better understanding
of the behavior of the Riemann zeta function. Moreover, Kevin A. Broughan and A.
Ross Barnett [37] considered the holomorphic flow of ζ. We emphasize that the Newton
flow is quite different to the holomorphic flow, since it involves the reciprocal of the
logarithmic derivative, the Newton quotient.
2.1. Continuous Newton method: essentials
The continuous Newton method is a steepest descent method with respect to an appropriately chosen gradient, the so-called Sobolev gradient [38–40]. The solutions s(t) of
the differential equation
ds(t)
F (s(t))
ṡ(t) ≡
=− 0
,
dt
F (s(t))
form the Newton flow of the complex function with derivative F 0 ≡ dF (s) . In appendix C,
ds
we elucidate the following properties:
(i) The Newton flow consists of lines with constant phase.
(ii) Zeros of F act like sinks for the Newton flow, whereas
(iii) poles are sources and
(iv) infinity can be both.
(v) The flow lines can only cross each other in hyperbolic points [41], where the derivative F 0 vanishes. We call these lines separatrices.
(vi) There are m separatrices crossing each other in the point s00 if the mth derivative
of F at s00 is the first non-vanishing derivative.
9
2. The Riemann zeta function: topology
Since the forward uniqueness of the trajectories gets lost in the hyperbolic point,
we refer to the different parts as incoming or outgoing separatrices according to their
direction with respect to the hyperbolic point. Incoming separatrices separate the flow
between neighboring zeros or poles, whereas outgoing separatrices divide flows with
different origin. Hence, the pictures of the Newton flow even reveal the location of
hyperbolic points.
2.2. Newton flow of the Riemann zeta function
Herein, we analyze the Newton flow of the Riemann zeta function. Figure 2.1 indicates
that its behavior in the neighborhood of the critical strip is dictated by the behavior
in the outer regions. Hence, we first explain the flow shown in the right and left panel
and make the connection to the central part from each side. Moreover, we restrict
our description to the upper half of the complex plane since the Newton flow of ζ is
symmetric with respect to the real axis.
2.2.1. Asymptotic for σ → +∞
Recalling the definition of the Dirichlet series, eq. (4.1), it becomes obvious that the
behavior of ζ for large positive real parts of s ≡ σ + iτ is governed by the function
Fe (s) ≡ 1 + e −s ln 2 ,
(2.1)
whose Newton flow is shown in appendix C.2.2. Indeed, the right panel of fig. 2.1 reveals
separatrices with vanishing phase (solid green lines) which come from left and arrive at
a zero of the derivative ζ 0 at σ → +∞ and τ = ±2k π/ ln 2, k ∈ N, marked by green
circles. The only curves which start at +∞ are the outgoing separatrices of these points.
They are located between the incoming ones at τ = ±(2k + 1) π/ ln 2 and indicated by
green arrows. The flow lines near the outgoing separatrices point to the right and turn
before they reach infinity, leading back to the critical strip in the neighborhood of the
outgoing separatrices.
When we examine the Newton flow at smaller values of σ, shown in the middle of
fig. 2.1 and for a larger range in fig. C.4, the structures become more complicated.
Nevertheless, the separatrices which continue in the right picture are clearly visible:
The incoming separatrices with imaginary part larger than π/ ln 2 ∼
= 4.5 cross the
complex plane from left to right forming ‘natural’ groups of non-trivial zeros. In contrast,
the outgoing separatrices with τ > 10 have to end in a non-trivial zero of ζ since they
alternate with the incoming separatrices. Moreover, it seems that all the curves coming
from right do not cross the critical line.
However, the separatrix at τ = π/ ln 2 and the flow around it acts differently. Although
the separatrix comes from the right and can be considered as outgoing part with respect
to the hyperbolic point at s00 = +∞, it is simultaneously an incoming separatrix for the
first trivial zero of ζ 0 at s00 ∼
= −2.72. It encloses the flow between the pole sp = 1 and
the first trivial zero s0 = −2.
10
2.2. Newton flow of the Riemann zeta function
Figure 2.1.: The Newton flow of ζ reveals the separatrices, indicated by green lines, and the pole
as well as the zeros and hyperbolic points, marked by black, red and green dots, respectively.
The behavior of the Newton flow in the central panel is governed by the asymptotic shown on the
left and right. From eq. (2.2) follows that the behavior of ζ for σ → −∞ is mainly determined
by χ, eq. (1.6). Hence, in the left panel, the separatrices are real with phase 0 (solid lines) or π
(dashed) in alternating sequence. They direct the flow into the trivial zeros of ζ 0 on the negative
real axis, which are slightly shifted to the right of the corresponding zeros of χ0 . We note, that
the behavior of the flow above the separatrix into the trivial zero of ζ 0 at σ ∼
= −4.94 is different
but not shown here: only some real positive lines are separatrices, the other flow lines end in the
non-trivial zeros of ζ.
In the right panel, the asymptotic for σ → +∞ is given by Fe , eq. (2.1). Even though all
separatrices are real and positive, only the incoming parts start at −∞, leading into the zeros
of ζ 0 at σ → +∞ and τ = 2k π/ ln 2, k ∈ N/{0}, indicated by green circles. These lines form
natural groups of the non-trivial zeros of ζ. The corresponding outgoing separatrices are the only
lines starting at +∞. Their initial imaginary parts is τ = (2k + 1) π/ ln 2, k ∈ N/{0, 1}, and they
end eventually in a non-trivial zero of ζ. However, the separatrix on the real line and the one
starting at s = +∞ + iπ/ ln 2 are special. They enclose the Newton flow from the pole sp = 1
into the first trivial zero s0 = −2, as the central panel shows. Finally, there are also non-real
separatrices determined by the non-trivial zeros of ζ 0 (dot-dashed green lines). Here, we have
only depicted the first one which separates the flow into the second and third non-trivial zero
of ζ. The phase on this line is ∼ 0.033. Hence, the curve is close to the real positive line (solid
violet). Moreover, the dashed violet lines, indicating ordinary flow lines with phase π, end in the
non-trivial zeros apparently without crossing the critical line (orange). In fig. C.4, we show the
Newton flow up to τ = 300.
11
2. The Riemann zeta function: topology
2.2.2. Asymptotic for σ → −∞
Now, we concentrate on the left panel of fig. 2.1. There, the source of the flow is at
σ → −∞, since (i) the flow from the only pole is restricted to disappear in the first
trivial zero, and (ii) all lines starting at +∞ are real separatrices which end in the
non-trivial zeros.
To explain the inclination of the flow lines with respect to the real axis, we make use
of the functional equation (1.5) and the approximation of the Dirichlet sum by Fe . This
yields
ζ(s) ∼
(2.2)
= 1 + e −(1−s) ln 2 χ(s)
showing that the asymptotic for σ → −∞ is mainly influenced by the function χ depicted
in fig. C.3. The trivial zeros of ζ and χ are the same and the trivial zeros of both functions
are located on the real axis, generating real separatrices with alternating signs, indicated
by dashed lines if the phase is equal to π. However, the position of the trivial zeros of ζ 0
and χ0 do not entirely coincide due to the influence of the Dirichlet series in eq. (2.2). Of
course, this influence gets weaker for σ → −∞, as shown in tab. C.1, and the pictures
of the flow look the same. A detailed analysis of the Newton flow of χ is given in
appendix C.2.1.
Figure 2.2 shows the differences between the separatrices and real lines of ζ and χ.
Here, we mark only the special points of the Riemann zeta function by dots: the pole
(black) and the zeros of ζ (red) and ζ 0 (green). The gray and blue lines indicate the
separatrices and real lines of χ, respectively, whereas the corresponding lines of ζ are
depicted in green and violet. The different styles of the curves represent the phases:
solid = 0, dashed = π and for the dot-dashed lines determined by the hyperbolic point
in which they cross. We neglect to draw the lines of χ to the right of the critical strip if
they are not necessary to describe the topology of ζ.
As expected, the real separatrices of χ through the hyperbolic points with σ < −4
are almost the same as the separatrices of ζ through the trivial zeros of ζ 0 . To make the
tiny deviations visible, we provide detailed views on the right side of fig. 2.2.
Moreover, the separatrix of χ with phase ∼ π/4 (dot-dashed gray curve) through its
non-trivial zero on the critical line at τ ∼
= 6.23 does not occur as separatrix in the Newton
flow of ζ. Below this line, χ possesses two real separatrices enclosing the flow between
its pole s = 3 and its zero s = −2 as well as the flow between its pole s = 1 and its zero
s = 0. In the case of the Riemann zeta function, we have only one pole at sp = 1 which
is connected by the flow to the first trivial zero s0 = −2.
Above the dot-dashed gray curve, the function χ has no separatrices. But in the
Newton flow of ζ some of the positive real lines of χ (solid blue curves) become the
positive real separatrices of ζ (green solid lines). They cross the critical strip and continue
as separatrices into the zeros of ζ 0 at +∞ familiar from the right panel in fig. 2.1. In
contrast, each negative real line ends at a non-trivial zero, apparently without crossing
the critical line. They are depicted as dashed violet curves.
12
2.2. Newton flow of the Riemann zeta function
Figure 2.2.: The separatrices of χ and ζ show that the influence of the Dirichlet series in the
approximation eq. (2.2) is quite small in the left half of the complex plane: For σ < −4, the
difference between the real separatrices of χ and ζ – depicted by gray and green lines, dashing
indicating the phase π – is only visible in the magnified views with higher resolution on the right.
In contrast, the separatrix through the non-trivial zero of χ on the critical line with τ ∼
= 6.23
(dot-dashed gray line) does not appear as separatrix in the flow of ζ, as well as other separatrices
to the right (see fig. C.3) which have been neglected in the graph. There are two real separatrices
of χ displayed below its non-trivial zero. They enclose the flow between the poles s = 1 and
s = 3 and the zeros s = 0 and s = −2 of χ. In contrast, the Riemann zeta function only possesses
one real separatrix in this region, since ζ has only one pole at sp = 1. This separatrix starts at
s00 = +∞ + iπ/ ln 2 and leads to the first trivial zero of ζ 0 at σ ∼
= −2.7. Above, we find different
separatrices of ζ, but only the non-trivial one of χ (dot-dashed gray line). The real separatrices of
ζ can be matched to some of the positive real lines of χ (solid blue curves), whereas the negative
real lines (dashed blue) are similar to the negative curves of ζ, indicated by dashed violet lines
which end in a non-trivial zero of ζ.
13
2. The Riemann zeta function: topology
2.2.3. Non-trivial separatrices
So far, we have only described the real separatrices of ζ determined by the trivial zeros
of ζ 0 located on the real axis or at σ → +∞. The outgoing parts of the last form natural
groups of two or more non-trivial zeros of ζ for τ > 18. Hence, the flow into different
zeros of one group must be separated by non-real curves through hyperbolic points.
This property of the Newton flow indicates that there are n − 1 non-trivial zeros of the
derivative ζ 0 in the region of a group with n non-trivial zeros of ζ, provided that the
higher derivatives of ζ do not vanish at the hyperbolic points. Otherwise, there are less
zeros of ζ 0 according to property (vi) listed in the section 2.1. Yet, fig. C.4 confirms that
up to τ = 300 no hyperbolic point exists where the second derivative of ζ vanishes [40].
Several results concerning the zeros of the derivatives ζ (k) are presented in [42].
Figure 2.1 only contains the separatrix through the first non-trivial zero of ζ 0
(dot-dashed green line), which is close to the real lines of ζ (violet curves) since its
phase is ∼ 0.033. We show in tab. C.2 that the phases of the non-trivial separatrices
up to imaginary part τ = 100 are quite small. This explains the ‘attraction’ of the real
lines by the hyperbolic points observed in [36].
Finally, we mark that the importance of the hyperbolic points becomes evident in the
formulation of the Riemann hypothesis by Speiser [35]:
The derivative ζ 0 has no zeros in the left half of the critical strip 0 < σ < 1/2.
Hence, it is worthwhile to further investigate the Newton flow of ζ and its derivatives,
since it would reveal if the Riemann hypothesis was violated.
14
3. Combined quantum systems
Our physical approach towards the Riemann zeta function takes advantage of its similarity to the time evolution of a quantum state. When we choose the associated Hamiltonian appropriately, we can reproduce the phases of the summands in the different
representations of ζ. The overlap with an adequate reference state then takes care of
the amplitudes. In this chapter, we introduce the general form of these states and show
how the phase space representations are connected to the overlap of the states.
3.1. Time evolution
We consider the quantum state
|Ψ(0)i ≡ |ψos i ⊗ |ψat i =
∞
X
n=0
ψn |ni ⊗ ce |ei + cg |gi ,
(3.1)
which is the direct product of the initial oscillator and atomic state, |ψos i and |ψat i,
respectively. The harmonic oscillator can be represented by a single mode of a cavity
field [43] or the motion of a trapped ion [44]. At time t = 0, it is given by a superposition
of Fock states |ni with probability amplitudes ψn . Likewise, the atomic state is in a
superposition of the ground and the excited state, |gi and |ei, of a two-level atom with
probability amplitudes cg and ce . Both states have to be normalized to ensure the
probability interpretation which implies
∞
X
n=0
|ψn |2 = 1
and
|ce |2 + |cg |2 = 1 .
The time evolution arises due to the Riemann Hamiltonian1
ĤR ≡ ~ω ln(n̂ + 1) σ̂z
(3.2)
in the interaction picture. Here, ω denotes the Rabi frequency which establishes the
coupling between the atom and the field mode and
σ̂z ≡ |eihe| − |gihg|
1
(3.3)
M. Planat et al. suspected in their paper [1] that it ‘may be that the logarithmic Hamiltonian [ln n̂]
is realized in the field of quantum physiology, where the perception states are proportional to the
logarithm of the excitation instead of being proportional to it.’
15
3. Combined quantum systems
is the Pauli spin matrix. The Hamiltonian ĤR is reminiscent of the effective Hamiltonian
of the Jaynes-Cummings-Paul model [28, 43]
ĤJC ≡ ~ω n̂ σ̂z ,
which is central to cavity quantum electrodynamics. Since this model has – despite of
its simplicity – a remarkable power of prediction, we give a short review and propose
some modifications to realize the Riemann Hamiltonian in chapter 8.
With the Hamiltonian ĤR , eq. (3.2), we get
|Ψ(t)i ≡ |ψe (t)i|ei + |ψg (t)i|gi
(3.4)
for the time-evolved state. It is crucial to keep in mind that the abbreviations
|ψe (t)i ≡ ce
∞
X
n=0
ψn e −iωt ln(n+1) |ni and |ψg (t)i ≡ cg
∞
X
n=0
ψn e +iωt ln(n+1) |ni
are in general not normalized to unity since the entangled state |Ψi has to be normalized.
Due to the fact that ĤR contains σ̂z , the two states accumulate the opposite phases
±iωt ln(n + 1) which depend linearly on time but logarithmically on the photon number
n. It is this superposition which creates the Riemann zeta function in the critical strip
as we see in what follows.
3.2. Joint measurement
We take the overlap
C(t) = hΦ|Ψ(t)i
between the time-evolved state |Ψ(t)i and the entangled reference state
X
|Φi ≡
|φa i|ai .
(3.5)
a
Here, a ∈ {e, g} and the in general unnormalized states |φa i are given by the expansion
|φa i ≡
∞
X
n=0
φna |ni
in Fock states with probability amplitudes φna . Equations (3.4) and (3.5) then yield the
expression
C(t) =
∞ X
n=0
φ ne ce ψn e −iωt ln(n+1) + φ ng cg ψn e iωt ln(n+1)
.
(3.6)
The dependence of the phase factor exp(−iωt ln(n + 1)) = (n + 1)−iωt indicates an
intimate connection between the overlap (3.6) and the Dirichlet representation (1.1)
16
3.3. Phase space representations
when we recall that the summands are n−σ−iτ . The term with the opposite phase is
reminiscent of the second sum in the Riemann-Siegel formula (1.9).
Hence, we use eq. (3.6) to produce the different representations of the Riemann zeta
function by quantum states in the next chapters. There, we also show that entanglement
is crucial to reach into the critical strip, but is unnecessary in the case of the Dirichlet
sum. Before we turn to the definition of the states, we give a short overview of the phase
space representations used afterwards to illustrate their behavior.
3.3. Phase space representations
For our purpose, the most suitable representation of a quantum state in phase space is
given by the Wigner formalism [18] and the closely related Moyal representation [20].
We give now a short overview of the required expressions and refer for a more detailed
description of the formalism and the properties of the functions to appendix D and to
literature, e.g. [28, 45].
3.3.1. Wigner and Moyal function
The well-known definition [18]
1
W|ψi (x, p) ≡
2π~
Z∞
i
ξ
ξ
dξ e − ~ pξ hx + |ψihψ|x − i
2
2
(3.7)
−∞
of the Wigner function provides us with the overlap
2
|hφ|ψi| = 2π~
Z∞
Z∞
dx
−∞
dp W|ψi (x, p) W|φi (x, p)
(3.8)
−∞
of two pure states |φi and |ψi. It involves the product of the corresponding Wigner
functions integrated over phase space, spanned by the position x and momentum p of
the wave function. Since the Wigner function is real per definition, eq. (3.8) makes only
the absolute value of a complex number available, but not its phase. Fortunately, we
can directly calculate the scalar product hφ|ψi with the expression
hφ|ψi =
Z∞
−∞
dx
Z∞
dp W|ψihφ| (x, p) ,
(3.9)
−∞
where we have used the generalize definition
1
W|ψihφ| (x, p) ≡
2π~
Z∞
i
ξ
ξ
dξ e − ~ pξ hx + |ψihφ|x − i
2
2
(3.10)
−∞
17
3. Combined quantum systems
of eq. (3.7) for the Moyal function of the two states. Equation (3.9) is quite powerful
since it yields the explicit formula
Z∞
Z∞
X
X
W|ψa ihφa | (x, p)
C = hΦ|Ψi =
hφa |ψa i =
dx
dp
a
−∞
−∞
(3.11)
a
for the scalar product of two entangled states |Φi and |Ψi of the form (3.5) in terms of
Moyal functions of their oscillator parts |φa i and |ψa i.
3.3.2. Wigner and Moyal matrix
Another representation of an entangled state in phase space is the Hermitian Wigner
matrix, eq. (D.9),
X
Ŵ|Ψi =
W|ψa ihψa0 | |aiha0 |
(3.12)
a,a0
with a, a0 ∈ {e, g}. This expression immediately follows from the definition of the Wigner
function, eq. (3.7), for a pure entangled state |Ψi of the form (3.5). In appendix D.3, we
give a generalization of this definition and its properties to arbitrary quantum states.
The Moyal functions W|ψa ihψa0 | are given in terms of the Wigner and Moyal functions
W|ni and W|nihm| , eqs. (D.20) and (D.17), of the Fock states by
"∞
#
∞ n−1
X
X
X
ψna ψ na0 W|ni +
W|ψa ihψa0 | =
ψna ψ ma0 W|nihm| + ψma ψ na0 W |nihm|
.
n=0
n=1 m=0
(3.13)
Due to the Hermiticity of the Wigner matrix, the diagonal elements
W|ψa ihψa | ≡ W|ψa i =
∞
X
n=0
|ψna |2 W|ni + 2
∞ n−1
X
X
n=1 m=0
Re ψna ψ ma W|nihm|
(3.14)
are the Wigner functions of the states |ψa i and therefore real. In contrast, the offdiagonal matrix elements are complex and connected via W|ψg ihψe | = W |ψe ihψg | . Moreover, the probability factors ψne and ψng merely scale the height of the Wigner and Moyal
functions of the Fock states since they are independent of the phase space variables x
and p.
As shown in appendix D.3, we find the probability amplitude
Z∞ Z∞
|C| ≡ |hΦ|Ψi| = 2π~ dx dp K
2
2
−∞
−∞
of two entangled states |Φi and |Ψi by integrating the kernel
X
K ≡ Tra Ŵ|Ψi Ŵ|Φi =
W|ψa ihψa0 | W|φa0 ihφa |
a,a0
18
(3.15)
(3.16)
3.3. Phase space representations
over the whole phase space. In analogy to eq. (3.8), K contains the product of the
corresponding Wigner representation (3.12) of the states. However, since Ŵ|Φi and Ŵ|Ψi
are operators, the product does not only include the Wigner functions W|ψa i ≡ W|ψa ihψa | ,
that is the diagonal elements of Ŵ|Ψi , but also the Moyal functions which represent the
off-diagonal matrix elements.
Just for the sake of completeness, we mention that the generalization of the Wigner
matrix (3.12) is obviously the Moyal matrix
X
Ŵ|ΨihΦ| =
W|ψa ihφa0 | |aiha0 |
a,a0
of the states |Φi and |Ψi, emerging directly form the definition of the Moyal functions
(3.10). This representation implies that the scalar product hΦ|Ψi, eq. (3.11), results
from the trace
X
W|ψa ihφa |
K ≡ Tra Ŵ|ΨihΦ| =
a
of the Moyal matrix integrated over phase space.
In the following chapters, we use the Moyal representation (3.11) and the Wigner
representation (3.15) of the overlap C(t), eq. (3.6), to describe the Riemann zeta function
in phase space.
19
4. Riemann states
In this chapter, we construct states which reproduce the Dirichlet series (1.1)
∞
X
1
ζ(s) ≡
ns
(4.1)
n=1
of the zeta function by a joint measurement. It is crucial to keep in mind that this
representation is only valid for Re s ≡ σ > 1, since this fact restricts the quantum states
to the same region. Therefore, we name these states Riemann states. Moreover, we
analyze the behavior of their phase space representations.
4.1. Definition and properties
When we rewrite the Dirichlet sum (4.1) by using s ≡ σ + iτ and shifting the summation
index, it becomes evident that
ζ(s) =
∞
X
n=0
1
e −iτ ln(n+1)
(n + 1)σ
(4.2)
holds some similarities with the quantum mechanical time evolution, eq. (3.4). Indeed,
choosing ce = 1 and cg = 0, the overlap C(t), eq. (3.6), between the time-evolved state
|Ψ(t)i and the entangled reference state |Φi yields
C(t) =
∞
X
φ ne ψn e −iωt ln(n+1) .
(4.3)
n=0
We can reproduce the Dirichlet representation (4.2) by eq. (4.3) if the probability amplitudes of the initial oscillator state and the reference state read
ψn (σ) ≡
N (σ)
(n + 1)σ/2
and
φne ≡ ψn (σ) ,
respectively. For the sake of simplicity, we choose φng = 0, since it does not contribute
to eq. (4.3). The overlap
C(t) = |N (σ)|2
∞
X
n=0
1
e −iωt ln(n+1) = |N (σ)|2 ζ(σ + iωt)
(n + 1)σ
21
4. Riemann states
emerges then from the scalar product hΦ|Ψi of the state
|Ψ(t)i = N (σ)
∞
X
1
e −iωt ln(n+1) |n, ei
σ/2
(n
+
1)
n=0
(4.4)
and its initial state |Φi ≡ |Ψ(0)i used as reference. The normalization is given by
!−1/2
∞
X
1
N (σ) =
= ζ −1/2 (σ)
(4.5)
(n + 1)σ
n=0
which is only convergent for σ > 1. Hence, with these states we can only describe the
zeta function in the region where the Dirichlet series (4.2) is convergent.
Equation (4.4) clearly shows that the time-evolved state |Ψ(t)i remains a product
state of the excited state |ei and the state
|σ, τ i ≡ N (σ)
∞
X
1
e −iτ ln(n+1) |ni
σ/2
(n
+
1)
n=0
at the rescaled time τ = ωt. Thus, the overlap
ζ(σ + iωt)
C(t) = he|hσ, 0| |σ, ωti|ei = hσ, 0|σ, ωti =
ζ(σ)
(4.6)
(4.7)
reduces to the scalar product of the time-evolved state |σ, ωti with its initial state,
revealing that entanglement is not necessary for the description of the zeta function in
the region σ > 1. Therefore, we call |σ, τ i Riemann state.
Before we turn to the analysis of the Wigner function of the Riemann state, it is
worthwhile to mention that |σ, τ i, eq. (4.6), is a generalization of the thermal phase
state [46]
∞
X
|ψp i ≡ Np
e −σn/2 |ni .
(4.8)
n=0
Both states are coherent superpositions of photon number states with real expansion
coefficients which decay with n. Their similarities and differences become evident, when
we compare the corresponding photon distributions: In the case of the thermal phase
state (4.8), we obtain
|hn|ψp i|2 = |Np |2 e −σn ,
(4.9)
which decays exponentially with the photon number n and has a maximum at n = 0,
that is for the vacuum state |0i.
For the Riemann state, the maximum of
|hn|σ, 0i|2 =
|N (σ)|2
= |N (σ)|2 e −σ ln(n+1)
(n + 1)σ
(4.10)
is also at n = 0. However, the photon statistics (4.10) only decays polynomially with
n, which is much slower than the exponential decay. In this sense, we have replaced n
by ln(n + 1) in eq. (4.9), in agreement with the construction of the Hamiltonian ĤR ,
eq. (3.2).
22
4.2. Wigner functions
Figure 4.1.: Wigner function1 W|σ,0i , eq. (4.11), of the initial Riemann state |σ, 0i for different
values of σ. For small values we get a tail aligned along the x-axis with interference fringes on
both sides created by the contribution of the Moyal functions. The interference fringes as well
as the tail vanish for larger σ due to the photon statistics, eq. (4.10).
4.2. Wigner functions
The definition of the Wigner function, eq. (3.7), yields for the Riemann state |σ, τ i the
expression
W|σ,τ i
n
o
n+1
−iτ ln m+1
∞ n−1
Re
e
W
X
X
|nihm|
W
1 
|ni
 ,
+2
=
σ/2
ζ(σ)
(n + 1)σ
[(n
+
1)(m
+
1)]
n=0
n=1 m=0

∞
X
(4.11)
where τ = ωt is the rescaled time. Of course, W|σ,τ i underlies the same restriction σ > 1
as the state |σ, τ i itself.
Like in the photon statistics (4.10), the Wigner function of the vacuum state |n = 0i
gives the largest contribution to the first sum. Although the Gauss shape of the Wigner
function W|0i is superimposed by the wells of the Wigner functions W|ni with larger n,
the whole sum remains positive in the entire phase space producing a structure which is
rotational symmetric with a maximum at the origin. For a more detailed description of
the shapes of W|ni we refer to appendix D.
Because of the factor [(n+1)(m+1)]−σ/2 in the double sum, the Moyal function W|nihm|
of the smallest energy eigenstates gives the highest contribution to this sum. For τ = 0,
the superposition of both sums produces a tail aligned along the x-axis which, due to
the denominator, gets longer for σ → 1 since more and more terms become significant
in the double sum. Figure 4.1 shows this characteristics.
Moreover, the Moyal functions shift the maximum of the first sum to the right and
produce interference fringes besides the tail. These properties become clear when we
examine the Wigner function of the so-called truncated Riemann state |σ, τ iν in section 5.2.
To understand the behavior of W|σ,τ i for τ > 0, we recall the explicit expressions of the
Wigner and Moyal functions of the Fock states, eqs. (D.20) and (D.21), which transform
1
All plots are created with Mathematica [47] and we use ~ = M = κ = 1 in all phase space plots.
23
4. Riemann states
Figure 4.2.: According to eq. (4.15) the Riemann zeta function given by the Dirichlet sum is
proportional to |C(t)|2 , depicted on top, which describes also the scalar product between the
initial Riemann state |σ, 0i and the time-evolved state |σ, ωti. In the Wigner formulation of
quantum mechanics this scalar product is given by the overlap in phase space between the
corresponding Wigner functions W|σ,τ i , eq. (4.11), presented in the middle row for σ = 2 and
three values of τ = ωt. The bottom row presents the product of these Wigner functions with
the Wigner function of the initial state W|2,0i shown in the left picture of fig. 4.1. Integration of
this product over phase space produces the values marked in the top picture by blue dots.
24
4.3. Moyal functions
eq. (4.11) into
W|σ,τ i
1 e −2|α|
=
ζ(σ) π~
2
"
∞
X
n=0
wnn
(n + 1)σ
#
wnm |α|n−m
n+1
.
+2
cos (n − m)β + τ ln
m+1
[(n + 1)(m + 1)] σ/2
n=1 m=0
∞ n−1
X
X
(4.12)
For the sake of simplicity, we have omitted the explicit notation of the dependence on
α = α(x, p), eq. (D.16), in its phase β ≡ arg α and in the real function wnm , eq. (D.18).
But we keep in mind that α is connected to the phase space variables via Re α ∼ x and
Im α ∼ p.
Equation (4.12) shows that the time-dependent factor in the double sum produces a
clockwise rotation of the Moyal functions
2
W|nihm|
e −2|α|
=
wnm |α|n−m e −i(n−m)β
π~
(4.13)
for n > m by the angle
n+1
.
(4.14)
m+1
This causes a rotation of W|σ,τ i for τ > 0 which results in a curling of the tail around the
origin, as depicted in the middle row of fig. 4.2 for σ = 2. Due to the different rotation
angles, eq. (4.14), the structure of the tail splits up and the initial symmetry in p gets lost.
ϕnm (τ ) ≡ τ ln
Finally, we make the connection between the Wigner functions of the states and their
overlap C(t), eq. (4.7), by recalling eq. (3.8), that is
Z∞ Z∞
ζ(σ + iωt) 2
.
|C(t)| = |hσ, 0|σ, ωti| = 2π~ dx dp W|σ, 0i W|σ,ωti = ζ(σ) 2
2
−∞
(4.15)
−∞
The bottom row of fig. 4.2 shows the product W|σ, 0i W|σ,τ i at different times τ = ωt
revealing that a small region of phase space around the central maximum produces the
main contribution to the overlap. The corresponding values of the overlap (4.15) are
marked by dots in the picture on top.
Since the Wigner formalism only produces the absolute value of ζ, we analyze in the
next section the representation of the scalar product hφ|ψi based on Moyal functions.
4.3. Moyal functions
The scalar product hσ, 0|σ, ωti of the initial and time-evolved Riemann state can be
evaluated directly by eq. (3.9), that is
Z∞ Z∞
ζ(σ + iωt)
C(t) = hσ, 0|σ, ωti = dx dp W|σ,ωtihσ, 0| =
,
ζ(σ)
−∞
(4.16)
−∞
25
4. Riemann states
Figure 4.3.: The lower pictures illustrate the time evolution of the Moyal function W|2,τ ih2,0| ,
eq. (4.17), in dependence on τ = ωt. The second row shows that the initial real part, which
is equal to W|σ,0i , becomes a short positive wedge with a negative ridge for τ = 1, while the
imaginary part displays a mainly negative wedge. For larger times, both wedges curl more and
more around the origin. The pictures of the absolute value (fourth row) are almost the same
as the ones of the real part since |ζ(2 + iτ )| ∼
= Re {ζ(2 + iτ )}, as the curves in the top picture
indicate. The values marked there by dots emerge from the Moyal functions by integration over
phase space according to eq. (4.16). The last row depicts the phase of W|2,τ ih2,0| which changes
between −π and π for τ > 0.
26
4.3. Moyal functions
in terms of the complex Moyal functions, eq. (3.10),
"∞
X e −iτ ln(n+1)
1
W|σ,τ ihσ,0| =
W|ni
ζ(σ)
(n + 1)σ
+
n=0
∞
X n−1
X
n=1 m=0
e −iτ
ln(n+1)
W|nihm| + e −iτ
ln(m+1)
[(n + 1)(m + 1)]σ/2
W |nihm|
#
(4.17)
providing us with the absolute value and the phase of ζ.
In contrast to the Wigner function W|σ,τ i of the Riemann zeta state |σ, τ i, eq. (4.11),
both sums in eq. (4.17) contain time-dependent factors and therefore produce complex
values for τ = ωt > 0.
Figure 4.3 illustrates the time evolution of the Moyal function W|σ,τ ihσ,0| for σ = 2.
Apart from the initial Moyal function, which is of course equal to the real-valued Wigner
function W|σ,0i depicted in the left picture of fig. 4.1, the real part of W|2,τ ih2,0| develops
a central maximum with a negative tail curling around the origin for τ > 0. In contrast,
the imaginary part possesses mainly a negative wedge at τ = 1, which curls in the same
way as the real part when time evolves.
The
shapes of absolute value |W|2,τ ih2,0| | are almost the same as the ones of
Re W|2,τ ih2,0| since |ζ(2 + iτ )| ∼
= Re {ζ(2 + iτ )}, confirmed by the upper picture. Of
course, the negative regions are positive in the case of the absolute value.
The bottom row depicts the phase of W|2,τ ih2,0| . The first picture only shows the
negative interference fringes of W|2,0i in blue, while for larger times the phase changes
around the tail from −π to π, indicated by the colors of the thermometer on the right.
Finally, when we integrate the Moyal functions according to eq. (4.16) over phase
space, we get the values of the corresponding part of ζ marked in the top picture by dots.
We conclude this chapter by reminding that the Riemann states are only suitable to
describe the Riemann zeta function for σ > 1. In the following chapters, we show that
we can penetrate the wall at σ = 1 with the help of the truncated Riemann states |σ, τ iν ,
which will be defined in section 5.2.
27
5. Alternating sum and truncated Riemann
states
One way to push the limitation σ > 1 to smaller values of σ is to use the alternating sum
representation (1.3) of the Riemann zeta function. In our physical picture, however, this
approach does not work. In this chapter, we show that all states which represent the
alternating sum exactly suffer from the same restriction σ > 1 as the Riemann states.
The only possibility to reach into the critical strip is to approximate ζ in this region,
which leads to the definition of truncated Riemann states.
5.1. Exact solution
A comparison between the overlap C(t), eq. (3.6), and the alternating sum of ζ, eq. (1.3),
yields the condition
∞
∞
X
X
Ma
(−1)n+1 −iτ ln(n+1)
!
ψn ce φ ne e −iτ ln(n+1) + cg φ ng e iτ ln(n+1) = 1−s
e
2
−1
(n + 1)σ
n=0
n=0
for the probability amplitudes. Here, we have already used the abbreviation τ = ωt. The
proportionality factor Ma contains the normalization constants of the states. Without
loss of generality, we choose real normalization constants from now on.
There are two possibilities to choose the probability amplitudes: (i) omitting entanglement by choosing cg = 0 like in the case of the Riemann state, or (ii) allowing
entanglement. We consider both ways separately in what follows.
5.1.1. Construction à la Riemann states
In the case of the Dirichlet sum, it is sufficient to consider the Riemann state which represents only the oscillatory system. Since the alternating sum is similar to the Dirichlet
representation, both describe ζ with one infinite sum, it is intuitive to choose the amplitudes
ce ≡ 1 ,
cg ≡ 0 ,
ψn ≡
N (σ)
(n + 1)σ/2
and φne ≡ N (σ)
(−1)n+1
.
(n + 1)σ/2
(5.1)
Here, the normalization N (σ) is given by eq. (4.5) which restricts the states again to the
region σ > 1. When we use other distributions of the phase (−1)n+1 in the probability
amplitudes, e.g.
ψn ≡ N (σ)
e iϕ(n+1)
(n + 1)σ/2
and φne ≡ N (σ)
e i(π−ϕ)(n+1)
(n + 1)σ/2
29
5. Alternating sum and truncated Riemann states
with ϕ ∈ [0, 2π), the same normalization occurs. Therefore, we cannot reach into the
critical strip with these states.
Now, the question arises if we can extend the region of definition by changing the
photon statistics of the states, that is using
(a)
ψn ≡
Nψ (σ)
with a > 0 .
(n + 1)σa
(5.2)
Indeed, for this state the normalization is given by
(a)
Nψ (σ)
≡
∞
X
n=0
1
(n + 1)2σa
!−1/2
= ζ −1/2 (2σa)
for σ >
1
2a
(5.3)
and we arrive at states with σ < 1 by choosing a > 1/2. However, the probability
amplitudes of the corresponding reference state have to be
(a)
φne ≡ Nφ
(−1)n+1
(n + 1)(1−a)σ
with the normalization
(a)
Nφ
≡
∞
X
1
(n + 1)2σ(1−a)
n=0
!−1/2
= ζ −1/2 (2σ(1 − a))
convergent for 2σ(1 − a) > 1. Hence, we need a < 1/2 for values σ < 1 which is not
(a)
allowed in the normalization Nψ , eq. (5.3). Only for a = 1/2 it is possible to get both
states normalized, but then σ has to be larger than 1 as we have shown before.
Needless to say, we can choose again other distributions of the phase (−1)n+1 in the
amplitudes of the state and reference state, but this does not change the normalizations.
5.1.2. Entanglement
It remains to show that even entanglement cannot√help to overcome the wall at σ = 1.
Indeed, when we choose for example ce = cg = 1/ 2 and the photon statistics ψn according to eq. (5.1), we can distribute the phases (−1)n+1 to both probability amplitudes
φne and φng of the reference state |Φi, eq. (3.5). For example, splitting the alternating
sum in an odd and an even part yields for the probability amplitudes
φne ∼
− δn,2k+1
(2k + 2)σ/2
and
φng ∼
δn,2k
(2k + 1)σ/2
with δn,k denoting the Kronecker delta. In this case, the normalization of the reference
state is again given by N (σ), eq. (4.5), which restricts the states to the plane with σ > 1.
Hence, the alternating sum is mathematically a powerful tool to reach into the critical
strip, but there are no physical states which could realize the infinite alternating sum
by the joint measurement described in section 3.2.
30
5.2. Truncated Riemann states
Figure 5.1.: The Wigner function of the truncated Riemann state |1/2, 0iν has ν − 1 minima
surrounding the central maximum which is shifted to the right of the origin. Both properties
result from the Moyal functions in the double sum of eq. (5.6). For larger summation limits ν,
the surrounding minima become a ring of interference fringes which faints further away from the
positive x-axis.
5.2. Truncated Riemann states
The simplest way to get normalizable states with probability amplitudes similar to
eq. (5.1) is to use only a finite number of Fock states. Therefore, we define in analogy to the Riemann state (4.6) the truncated Riemann state
|σ, τ iν ≡ Nν (σ)
ν−1
X
1
e −iτ ln(n+1) |ni
σ/2
(n + 1)
n=0
with its normalization
Nν (σ) ≡
ν−1
X
n=0
1
(n + 1)σ
!−1/2
(5.4)
(5.5)
valid for all σ > 0 if ν is finite. Such truncated states can be realized with methods of
state engineering [48] and state synthesis [49].
For ν → ∞ the expression (5.4) becomes equal to the definition (4.6) of the Riemann
state |σ, τ i∞ ≡ |σ, τ i and, due to the normalization, again restricted to σ > 1. Nevertheless, the truncated Riemann states provide a deeper insight into the structure of the
Riemann states as we show now.
31
5. Alternating sum and truncated Riemann states
The Wigner function of the truncated Riemann state
n
o
n+1
−iτ ln m+1
ν−1 n−1
Re
e
W
X
X
|nihm|
W
|ni

+2
W|σ,τ iν (α) ≡ Nν2 (σ) 
σ/2
(n + 1)σ
[(n
+
1)(m
+
1)]
n=1 m=0
n=0

ν−1
X
(5.6)
is the truncated version of W|σ,τ i , eq. (4.11). It is obvious that the photon statistics
|hn|σ, τ iν |2 =
Nν2 (σ)
(n + 1)σ
for
n<ν
enhances the contributions of the Fock states with small photon number. Hence, the
highest contribution to the first sum in eq. (5.6) is given by the Wigner function of the
vacuum state W|0i , which is positive in the whole phase space with a maximum at the
origin. The contributions of the higher Fock states do not change these properties. Even
the sum over the Wigner functions without the factors (n + 1)−σ is positive, as we show
in appendix D.4.1.
In the double sum the Moyal function W|1ih0| gives the largest contribution, which
– due to the shape of its real part – enhances the values at x > 0 and reduces them
for x < 0 when superposed with the first sum. This becomes evident in the second
picture of fig. 5.1 where W|1ih0| is the only contribution of the double sum. The real
parts of the higher Moyal functions are always zero at the origin and the contributions
with odd m have a minimum following on the positive x-axis (see fig. D.2). Hence, the
maximum of the first sum at the origin gets shifted to the right as we see in the other
pictures of fig. 5.1. Since W|ni and W|nihm| , eqs. (D.20) and (D.21), contain the factor
2
e −2|α| , acting as an envelop, the whole structure flattens with
increasing distance to the
origin. Moreover, the star-shaped patterns of Re W|nihm| , depicted in fig. D.2, lead to
ν − 1 minima around the origin. They create for larger summation limits ν a ring with
increasing radius in which the structure distant to the positive x-axis faints. Thus, they
develop the tail aligned along the x-axis which is known from the Wigner function of the
Riemann state |σ, τ i. Although the structure of the truncated Riemann state resembles
the one of |σ, τ i, we have to keep in mind that, due to the normalization, they can only
become equal for ν → ∞ if σ > 1.
For times τ 6= 0 the real parts of the Moyal functions W|nihm| are additionally rotated
clockwise around the origin with angles ϕnm (τ ) given by eq. (4.14). Since the rotation
angle changes for different combinations of n and m the initial symmetry in p gets
destroyed. Hence, the whole structure of the Wigner function W|σ,τ iν becomes even
more complicated for large τ and, of course, with increasing ν. The left pictures in the
bottom part of fig. 5.2 show this behavior for W|2/3,τ i100 at different times τ .
We postpone the description of the other pictures shown in fig. 5.2 to section 5.3.2
and analyze the influence of the truncation on the alternating sum before.
32
5.3. Truncated alternating sum
Figure 5.2.: The absolute value squared of the overlap C, eq. (5.8), which gives the values of
the truncated alternating sum, is shown in blue in the upper right picture for σ = 2/3 and
(100) 2
ν = N = 100. The red dashed line indicates |Ma
ζ| , emphasizing the good agreement
between the approximation (5.8) and the exact values for large N . The values marked by dots
can be calculated from the products W|φiN W|σ,τ iN , depicted below in the right pictures, by
integration over phase space according to eq. (5.12). In the pictures of the products the values
at different points are much smaller than the ones of the Wigner functions W|σ,τ iN , eq. (5.6),
shown on the left. The clockwise rotation of the Wigner function curls the tail around the origin.
The interference fringes besides the tail fade away with larger distance to the x-axis. The upper
left picture confirms that W|φiN is W|σ,0iN mirrored on the p-axis, in agreement with eq. (5.11).
33
5. Alternating sum and truncated Riemann states
Figure 5.3.: The absolute value of the truncated alternating sum ζN , eq. (5.7), is depicted for
σ = 1/2 (red) and σ = 2/3 (blue) for the truncation parameters N = 100 and N = 200 in the
first row. As expected, for both N the red line is close to zero at the zeros of ζ, marked by
the vertical dashed lines at τ ∼
= 14.135, τ ∼
= 21.022, τ ∼
= 25.011 and τ ∼
= 30.425. However, the
magnified pictures of the difference |ζN | − |ζ| in the bottom show that the values do not vanish
entirely at these points. Nevertheless, the quality of the approximation ζN increases for larger
N as well as for increasing σ, as the pictures of the difference in the second row confirm.
34
5.3. Truncated alternating sum
σ = 1/2
σ = 2/3
|ζN |
6
4
2
680
682
684
680
682
684
τ
Figure 5.4.: The absolute value of the truncated alternating sum ζN , eq. (5.7), is depicted for
σ = 1/2 (red) and σ = 2/3 (blue) for the truncation parameters N = 200 and N = 230 by
dashed and dotted lines, respectively. The solid lines indicate the exact curves ζ ≡ ζ∞ . The
pictures show that we need higher truncation parameters N for larger imaginary parts τ to get
an adequate approximation of ζ ≡ ζ∞ . The dashed vertical lines indicate the zeros of ζ at
τ∼
= 681.895, τ ∼
= 682.603 and τ ∼
= 684.014.
= 679.742, τ ∼
5.3. Truncated alternating sum
In analogy to the truncated Riemann states, we define the truncated alternating sum
ζN (s) ≡
1
21−s − 1
N
−1
X
n=0
(−1)n+1
(n + 1)s
(5.7)
by cutting the sum in eq. (1.3) at a finite truncation parameter N . Figures 5.3 and
5.4 show that the quality of the approximation ζN strongly depends on the chosen
parameters and can always be improved by enlarging the truncation N .
The first row in fig. 5.3 presents the absolute value of ζN for σ = 1/2 (red) and σ = 2/3
(blue) as well as N = 100 and N = 200 for small imaginary parts τ . As expected, the
red curve is close to zero at the zeros of ζ, marked by the vertical dashed lines. However,
the magnified pictures of the difference |ζN | − |ζ| in the bottom rows confirm that the
values do not vanish entirely at these points. The zeros are only reached in the limit
N → ∞. For σ = 2/3, the difference to the exact values is smaller, since the influence
of the summands in eq. (1.3) with large n fades for increasing σ.
Figure 5.4 demonstrates that we need larger truncation parameters N for larger imaginary parts τ to get a reliable approximation of the Riemann zeta function ζ ≡ ζ∞ ,
indicated by the solid curves. The dashed and dotted lines represent N = 200 and
N = 230, respectively.
In what follows, we use σ = 2/3 in the phase-space pictures of the states and the
overlap since the deviation of ζN to the exact values is smaller than for σ = 1/2. The
description of ζ for smaller σ is subject to chapter 6 and 7, where we reproduce the
Riemann-Siegel and Berry-Keating formula which are more suited to approximate ζ in
the critical strip.
35
5. Alternating sum and truncated Riemann states
5.3.1. Reference state and overlap
Now, we turn to the calculation of the overlap
)
C(t) = N hφ|σ, ωtiν = M(ν,N
(σ, ωt) ζN (σ + iωt) .
a
(5.8)
With the definition (5.4) of the truncated Riemann state |σ, τ iν and the truncated alternating sum, eq. (5.7), we find the reference state
|φiN ≡ NN (σ)
N
−1
X
n=0
(−1)n+1
|ni
(n + 1)σ/2
(5.9)
with N ≤ ν and the proportionality factor
)
M(ν,N
(σ, τ ) ≡ NN (σ) Nν (σ) 21−σ−iτ − 1 .
a
(5.10)
In appendix F.1, we show that the normalization NN , eq. (5.5), decreases for increasing
truncation limit N and σ in the critical strip. This behavior makes the proportionality
(N )
(N,N )
factor Ma ≡ Ma
, analyzed in appendix F.4, as well as the overlap C small for
large N . Nevertheless, we examine the Wigner and Moyal representations of ζN in the
remaining section.
5.3.2. Wigner representation
Equation (3.14) yields the Wigner function
W|φiN = NN2 (σ)
" N −1
X
n=0
#
N
−1 n−1
X
X (−1)n+m Re W|nihm|
W|ni
+2
(n + 1) σ
[(n + 1)(m + 1)] σ/2
n=1 m=0
of the reference state |φiN defined by eq. (5.9). This expression is reminiscent of the
Wigner function W|σ,0iN of the initial truncated Riemann state. Indeed, the first part is
for ν = N the same as in eq. (5.6) and therefore symmetric in x and p. With the help
of eq. (4.13), we can transform the numerator in the double sum into
n
o
(−1)n+m Re W|nihm| (α) = fnm (|α|) Re e −i(n−m)(β+π) = Re W|nihm| (−ᾱ) .
Here, the abbreviation fnm (|α|) denotes the absolute value of W|nihm| , which only depends on |α|. When we now use the p-symmetry of the Moyal functions and the symmetries of the first sum, we see that the Wigner function of the reference state is given
by
W|φiN (x, p) = W|σ,0iN (−ᾱ) = W|σ,0iN (−x, p) .
(5.11)
Hence, W|φiN is the opposite hand of the Wigner function of the initial truncated Riemann state |σ, 0iN with respect to the p-axis, as the upper left picture of fig. 5.2 confirms.
36
5.3. Truncated alternating sum
Finally, we can represent the overlap (5.8) in terms of the Wigner functions by the
expression
2
2
|C(t)| = |N hφ|σ, ωtiν | = 2π~
Z∞
−∞
dx
Z∞
dp W|σ,ωtiν (x, p) W|σ,0iN (−x, p) .
(5.12)
−∞
It is depicted in blue in the upper right picture of fig. 5.2 for τ = ωt. The red dashed
(100)
line indicates |Ma (2/3, τ ) ζ(2/3 + iτ )|2 to emphasize the good agreement between
the approximation by the truncated sum ζ100 and the exact values of ζ.
The right pictures below show the product W|φiN W|σ,τ iN of the Wigner functions for
three different times. They have a central structure which is shifted to the left due to
the mirrored shape of W|φiN . The surrounding interference fringes are created by the
tails of the Wigner functions. When we integrate the data over phase space according
to eq. (5.12), we get the values of |C|2 marked in the upper right picture by dots.
5.3.3. Moyal representation
According to eq. (3.9), the overlap C(t) = N hφ|σ, ωtiν , eq. (5.8), is equal to the Moyal
function, eq. (3.13),
"N −1
X (−1)n+1 e −iτ ln(n+1) W|ni
W|σ,τ iν N hφ| = Nν (σ) NN (σ)
(n + 1)σ
n=0
#
N
−1 n−1
X
X (−1)m+1 e −iτ ln(n+1) W|nihm| + (−1)n+1 e −iτ ln(m+1) W |nihm|
+
[(n + 1)(m + 1)]σ/2
n=1 m=0
(5.13)
of the truncated Riemann state |σ, τ iν at τ = ωt and the reference state |φiN integrated
over the whole phase space. Here, the choice N ≤ ν again truncates the sums over n at
N − 1. As in the case of the Riemann states, both sums of the Moyal function contain
time-dependent factors, but now additionally multiplied by a phase factor ±1.
We show in the bottom of fig. 5.5 the real and imaginary part as well as the absolute
value of the Moyal function W|σ,τ iν hφN | for σ = 2/3, N = ν = 100 and different values of
τ . The shapes for τ = 0 are all symmetric in x and additionally symmetric in p in the
case of the real part and the absolute value, while the imaginary part is antisymmetric in
p. For larger times, these symmetries get lost since the truncated Riemann state |σ, τ iν
starts to rotate.
The picture on top presents the dependence of the overlap C(t), eq. (5.8), on τ . The
values marked by dots emerge from the pictures by integration over phase space as
defined in eq. (3.9).
37
5. Alternating sum and truncated Riemann states
Figure 5.5.: In the bottom pictures the Moyal function W|2/3,τ i100 hφ| , eq. (5.13), is depicted for
ν = N = 100 and different values of τ by its real and imaginary part as well as by its absolute
value. For τ = 0, we find structures which are symmetric in x and in the case of the real part
and absolute value additionally symmetric in p, while the initial imaginary part is antisymmetric
in p. All symmetries get lost for τ > 0 due to the rotation of the truncated Riemann state
|2/3, τ i100 . To make the main structure more visible, we have cut the outer regions where only
small interference fringes occur. Integration of W|2/3,τ i100 hφ| over phase space gives the values
marked in the upper picture due to eq. (3.9) which connects the overlap of the states with the
Moyal representation.
38
5.4. Summary
5.4. Summary
This chapter has demonstrated that the exact version of the alternating sum can only
be produced by quantum states with σ > 1. Indeed, if we use only one quantum system,
the normalization of the states forbids photon statistics with σ ≤ 1. Even combinations
of two entangled states fail, since one of them is, due to its normalization, restricted to
the same region.
To circumvent this problem, we have truncated the alternating sum. The resulting
states are even for σ ≤ 1 normalizable. However, the quality of the approximation with
the truncated alternating sum ζN depends strongly on the used parameters. Especially
in the critical strip, we need for large imaginary parts τ large truncation limits N to get
adequate results. But large values of N produce small normalization constants for the
(ν,N )
states which reduce the value of the proportionality factor Ma
and therefore make
the overlap C tend to zero.
Hence, we dedicate the next chapters to the investigation of the more suitable approximations of ζ to enter the critical strip: the Riemann-Siegel and Berry-Keating formula.
39
6. Riemann-Siegel states
In the previous chapters, we have shown that the Dirichlet representation of the Riemann
zeta function leads to states only normalizable for σ > 1. Even the states which produce
the alternating sum exactly suffer from the same restriction in the physical picture.
Only the truncated Riemann states are able to enter the critical strip. However, the
accuracy of the corresponding approximation of ζ by the truncated alternating sum
strongly depends on the chosen parameters. For large imaginary parts τ , we need quite
large truncation limits N , which result in small probability amplitudes and therefore in
a very small overlap of the states. Besides, the functional equation (1.5) is of no use
either, since the above expressions only expand the region to σ < 0, missing again the
critical strip.
Hence, we employ now the Riemann-Siegel formula ζRS which provides a good approximation of ζ in the critical strip, especially for large imaginary parts τ . In contrast to the
alternating sum, the summation limit n0 is quite small and consequently, the resulting
amplitudes of the states are much larger. Yet, we will see that the states have to be
entangled to reproduce the correct phase dependence of the summands in ζRS.
In the following, we restrict our investigation to positive σ. The values of ζ for negative
σ can be found with the functional equation (1.5).
6.1. Definition of the states
We start from the main part of the Riemann-Siegel formula [29]
n0
n0
X
X
1
1
ζRS(s) ≡
+ χ(s)
,
s
1−s
n
n
n=1
(6.1)
n=1
where χ(s) and n0 are defined by eqs. (1.6) and (1.10), respectively. When we use again
s = σ + iτ and shift the index of the sum in eq. (6.1), we arrive at
ζRS(σ + iτ ) =
nX
0 −1
n=0
"
e iτ ln(n+1)
e −iτ ln(n+1)
+
χ(σ
+
iτ
)
(n + 1)σ
(n + 1)1−σ
#
.
(6.2)
The second sum accumulates phases with opposite sign with respect to the first sum.
As we mentioned in chapter 3, the same combination of phases occurs in the JaynesCummings-Paul model which describes the interaction of a two-level atom with a single
mode of the cavity field. Following this example, we can construct the states producing
the Riemann-Siegel formula.
41
6. Riemann-Siegel states
6.1.1. Riemann-Siegel state
First, we choose as oscillator state the truncated Riemann state |σ, τ iν , eq. (5.4), and√an
equal distribution between the atomic probability amplitudes, that is ce = cg = 1/ 2.
The initial state then reads
1
|ΨRS(σ, 0)i ≡ |σ, 0iν ⊗ √ |ei + |gi
2
according to eq. (3.1). Time evolution with ĤR yields the state
1
|ΨRS(σ, τ )i ≡ √ |σ, τ iν |ei + |σ, −τ iν |gi
2
(6.3)
which consists of two counter-rotating truncated Riemann states, |σ, τ iν and |σ, −τ iν ,
each entangled to one part of the two-level atom. Since this state contains the desired
phases ±iτ ln(n + 1), we call it Riemann-Siegel state. Our next task is to choose the
appropriate reference state.
6.1.2. Riemann-Siegel reference state
Now, we turn to the examination of the scalar product
CRS(t) ≡ hΦRS|ΨRS(σ, ωt)i = MRS ζRS(σ + iωt)
(6.4)
to identify the probability amplitudes of the reference state |ΦRSi defined by eq. (3.5).
The proportionality factor MRS contains the normalizations of the states. With ζRS and
|ΨRSi given by eqs. (6.2) and (6.3), respectively, we find
ν−1 Nν (σ) X
CRS(t) = √
2 n=0
= MRS
φ ne
n0 (τ )−1 X
n=0
1
1
e −iωt ln(n+1) + φ ng
e iωt ln(n+1)
σ/2
(n + 1)
(n + 1)σ/2
1
1
e −iτ ln(n+1) + χ(σ + iτ )
e iτ ln(n+1)
σ
(n + 1)
(n + 1)1−σ
. (6.5)
At first, we note that the range of the summation index n is different on both sides of
eq. (6.5). The sum on the left-hand side can only become equal to the one on the right
if
n0 (τ ) ≤ ν ,
(6.6)
since then we can choose for the probability amplitudes of the reference state
φne = φng = 0
for
n ≥ n0 ,
which truncates the sum on the left at n0 . It is crucial to keep in mind that the right-hand
side of eq. (6.5) and, therefore, especially n0 depends on τ ≡ Im s.
Then, we compare the exponential functions on both sides and find additionally the
condition τ = ωt, which, in contrast to the previous chapters, cannot be identified as
42
6.2. Entanglement
a rescaled time. Indeed, due to eq. (6.6), the identification of τ as rescaled time would
imply that the truncation parameter ν of the Riemann-Siegel state is time-dependent.
However, eq. (6.6) only requires that ν is not smaller than the constant n0 chosen for
the parameter τ ≡ Im {σ + iωt}.
Finally, we find that the reference state
|ΦRS(σ, τ )i = Nn0 (σ, τ ) |σ, 0in0 |ei + γ (σ, τ ) |2 − 3σ, 0in0 |gi
(6.7)
with
fulfills eq. (6.4) if
1
Nn0 (σ, τ ) ≡ q
1 + |γ(σ, τ )|2
γ(σ, τ ) ≡
and thus
MRS ≡
(6.8)
Nn0 (σ)
χ(σ + iτ )
Nn0 (2 − 3σ)
(6.9)
Nν (σ)
√
Nn0 (σ) Nn0 (σ, τ ) .
2
6.2. Entanglement
Before we turn to the investigation of the Wigner matrix elements, we analyze if the
states |ΨRSi and |ΦRSi are entangled as we assume by their construction. The methods
of testing for entanglement are described in appendix E.
6.2.1. Riemann-Siegel state
For the Riemann-Siegel state, eq. (6.3), the probability amplitudes fulfill ψng = ψ ne ,
which simplifies eq. (E.7) to
2
ν−1
X
2 ψne
≤
n=0
ν−1
X
n=0
|ψne |2
!2
.
Moreover, ψne is of the form rn e −iθn leading to
ν−1
ν−1
X
X
2 −2iθn ≤
rn2 .
r
e
n
n=0
n=0
Hence, the sum on the left-hand side can only be the same as the right one if the phases
fulfill
!
θn ≡ τ ln(n + 1) = 2πk
with
k∈N
∀ n = 0, ..., ν .
This is of course the case for τ = 0, since the initial state is a product state, or if ν = 1,
but it cannot be fulfilled for ν ≥ 2 and τ 6= 0 for all n at once.
43
6. Riemann-Siegel states
6.2.2. Riemann-Siegel reference state
Now, we use the overlap of the field states, eq. (E.3), to analyze the entanglement of the
reference state |ΦRSi directly. From representation (6.7) follows that
2
nX
0 −1
1
|n0 hσ, 0|2 − 3σ, 0in0 |2 = Nn0 (σ) Nn0 (2 − 3σ)
≤1
(n + 1)1−σ n=0
or
nX
0 −1
n=0
1
(n + 1)1−σ
!2
≤
nX
0 −1
n=0
nX
0 −1
1
1
σ
(n + 1)
(m + 1)2−3σ
(6.10)
m=0
when we use the definition (5.5) of the normalization Nn0 . In appendix E.4, we show
that for n0 ≥ 2 equality in eq. (6.10) is only given if σ = 1/2. Hence, the reference state
|ΦRSi is only for σ = 1/2 disentangled, hinting the peculiarity of the critical line.
We note that we can also investigate the entanglement by determining the Schmidt
decomposition of the density matrix ρ̂ ≡ |ΦRSihΦRS| as described in appendix E.2. However, there is no obvious connection between the eigenvalues of the reference state and
the location of the zeros of the Riemann zeta function.
6.3. Wigner matrix of the Riemann-Siegel state
As we have seen in section 3.3.2, the Wigner representation of an entangled state is given
by the Hermitian Wigner matrix, eq. (3.12), which reads for the Riemann-Siegel state,
eq. (6.3),
1
Ŵ|ΨRSi ≡
W|σ,τ iν |eihe| + W|σ,−τ iν |gihg| + W|σ,τ iν hσ,−τ | |eihg| + W|σ,−τ iν hσ,τ | |gihe| .
2
(6.11)
Hence, the diagonal matrix elements are given by the Wigner functions of the truncated
Riemann state divided by two, shown in fig. 5.1 and on the left in fig. 5.2.
As expected, the diagonal matrix element connected to the excited state |ei rotates
clockwise whereas the one connected to the ground state |gi rotates counter-clockwise
in phase space according to
W|σ,−τ iν (x, p) = W|σ,τ iν (x, −p) .
(6.12)
This behavior is shown in the first and second row of fig. 6.1 for σ = 1/2 and ν = 10.
Needless to say, for τ = 0 the diagonal matrix elements are equal to each other and
symmetric in p.
Now, we turn to the off-diagonal matrix elements in eq. (6.11) which are given by the
44
6.3. Wigner matrix of the Riemann-Siegel state
Figure 6.1.: Time evolution manifests itself in the matrix elements of the Wigner matrix Ŵ|ΨRS i
of the Riemann-Siegel state in different ways. Caused by the rotation of the corresponding
Wigner function W|σ,±τ iν of the truncated Riemann state, the diagonal matrix elements, shown
in the two upper rows, rotate clockwise or counter-clockwise. In contrast, the off-diagonal matrix
element, which is proportional to the Moyal function W|σ,τ iν hσ,−τ | , remains symmetric in p but
changes its inner structure. The pictures display the matrix elements for σ = 1/2 and ν = 10. At
τ = 0, all matrix elements are the same and proportional to W|1/2,0i10 already shown in fig. 5.1.
45
6. Riemann-Siegel states
Moyal function
W|σ,τ iν hσ,−τ | = Nν2 (σ)
" ν−1
X e −2iτ ln(n+1)
(n + 1) σ
W|ni
n=0
ν−1
X n−1
X
#
e −iτ ln[(n+1)(m+1)]
Re W|nihm| .
+2
[(n + 1)(m + 1)] σ/2
n=1 m=0
(6.13)
In contrast to the Wigner functions W|σ,τ iν , eq. (5.6), the time-dependence acts in both
sums as factor and the Moyal functions of the Fock states are not rotated. Hence, the
initial symmetry of W|σ,0iν hσ,0| ≡ W|σ,0iν in p is conserved for all times τ , that is
W|σ,τ iν hσ,−τ | (x, −p) = W|σ,τ iν hσ,−τ | (x, p) .
(6.14)
Moreover, for τ 6= 0 and ν > 1, the complex factors in both sums suppress or even
produce negative contributions of W|ni and W|nihm| to the real or imaginary part of
the Moyal function W|σ,τ iν hσ,−τ | . This leads to different distributions of positive and
negative regions in the patterns shown in the two bottom rows of fig. 6.1 compared to
the structures of the Wigner functions depicted above. Here, σ = 1/2 and ν = 10.
It remains to mention that for larger values of σ the internal structures of the patterns
of the Wigner matrix elements are less distinct since, due to the photon statistics, the
contributions for higher n are smaller.
6.4. Wigner matrix of the reference state
The Wigner matrix, eq. (3.12), of the Riemann-Siegel reference state |ΦRSi, eq. (6.7), is
given by
Ŵ|ΦRSi ≡ Nn20 W|σ,0in0 |eihe| + |γ|2 W|2−3σ,0in0 |gihg|
+γ W|σ,0in0 h2−3σ,0| |eihg| + γ W|2−3σ,0in0 hσ,0| |gihe| .
(6.15)
Since the diagonal matrix elements are proportional to the Wigner functions of the
initial truncated Riemann states |σ, 0in0 and |2 − 3σ, 0in0 , eq. (4.11), they are always
symmetric in p. Moreover, the shapes of W|σ,0in0 and W|2−3σ,0in0 differ only in height as
long as the condition n0 (τ1 ) = n0 (τ2 ) is fulfilled for different values τ1 and τ2 , because
then the dependence on τ only appears in the factors Nn20 and |γ|2 Nn20 .
The two upper rows of fig. 6.2 show the diagonal matrix elements of Ŵ|ΦRSi for τ = 700
(n0 = 10) at σ = 1/3 and σ = 1/2. Due to |γ| = 1 for σ = 1/2, the diagonal matrix
elements are the same on the critical line. In contrast, the structures for σ = 1/3 are
only similar to each other. Although the values of the Wigner function W|1,0in0 are
smaller than the ones of W|1/3,0in0 , the matrix element connected to the ground state
|gi is more pronounces because of the larger factor |γ|2 Nn20 . A detailed analysis of the
normalization factors is given in appendix F.5.
46
6.4. Wigner matrix of the reference state
Figure 6.2.: Wigner matrix elements of the Riemann-Siegel reference state at τ = 700 (n0 = 10)
for σ = 1/3 and σ = 1/2. The patterns of the diagonal matrix elements depicted in the first
and second row are always symmetric in p. For each σ, their shapes change only in height if
n0 is the same for different τ . However, due to the factor |γ|2 Nn20 , the diagonal matrix element
connected to the ground state is more pronounced for σ < 1/2 than the one connected to the
excited state. The pictures in the two bottom rows show the real and imaginary part of the
off-diagonal Wigner matrix element which is connected to the Moyal functions W|σ,0in0 h2−3σ,0| .
The patterns are only on the critical line symmetric in p. For σ = 1/3 they are nearly symmetric
or anti-symmetric, respectively, since the phase of γ is equal to arg χ ≈ 0.89 π.
47
6. Riemann-Siegel states
The two bottom rows of fig. 6.2 present the real and imaginary part of the off-diagonal
matrix elements of Ŵ|ΦRSi , which are connected to the Moyal functions, eq. (3.10),
W|σ,0in0 h2−3σ,0| ≡ Nn0 (σ) Nn0 (2 − 3σ)
nX
0 −1
W|nihm|
(n + 1)σ/2 (m + 1)(2−3σ)/2
n,m=0
(6.16)
via the factor |γ| e ±i arg γ Nn20 . Hence, on the critical line they are – apart from the phase
of the factor – the same as the diagonal matrix elements and therefore symmetric in p.
Thus, all matrix elements differ for σ = 1/2 only in height but not in shape.
However, for σ 6= 1/2, the shapes of the off-diagonal matrix elements have in general
no symmetry in p since the phase ± arg γ = ± arg[χ(σ + iτ )], defined by eq. (6.9), rotates
the Moyal functions W|nihm| of the Fock states. Only for the special case arg χ = πk with
k ∈ Z, the real part of the off-diagonal matrix elements is symmetric in p whereas the
imaginary part is antisymmetric. These properties are reversed for arg χ = (2k + 1)π/2.
Since arg χ(1/3 + 700 i) ≈ 0.89 π, the bottom pictures of fig. 6.2 for σ = 1/3 are nearly
symmetric or antisymmetric, respectively.
6.5. Overlap in Wigner representation
Now, we have all ingredients to determine the overlap
Z∞ Z∞
2
|CRS| = 2π~ dx dp KRS = MRS
|ζRS(σ + iωt)|2
2
−∞
(6.17)
−∞
in Wigner representation from eq. (3.15) and, with the help of eq. (6.4), the absolute
value of the Riemann-Siegel representation ζRS. The kernel, eq. (3.16),
KRS ≡
Nn20 h
W|σ,τ iν W|σ,0in0 + |γ|2 W|σ,−τ iν W|2−3σ,0in0
2
n
oi
+ 2 Re γ W|σ,τ iν hσ,−τ | W |σ,0in0 h2−3σ,0|
(6.18)
follows from the definitions (6.3) and (6.7) of the Riemann-Siegel state and its reference
state. Here, we have combined the off-diagonal matrix elements using that the Wigner
matrix is Hermitian.
A closer look on KRS reveals that it is only on the critical strip symmetric in p. This
becomes evident when we recall the properties of the matrix elements of the RiemannSiegel state and its reference state:
The diagonal matrix elements of the Riemann-Siegel state are connected via the relation (6.12) while the diagonal matrix elements of the reference state are symmetric in
p and even the same for σ = 1/2. Since the prefactors are also the same on the critical
line, the sum in the first row of eq. (6.18) is symmetric in p for σ = 1/2.
The off-diagonal matrix elements of the Riemann-Siegel state are always symmetric
in p, eq. (6.14), whereas this holds only for σ = 1/2 in the case of the reference state.
Hence, KRS is only on the critical line symmetric in p.
48
6.5. Overlap in Wigner representation
Figure 6.3.: The upper pictures show |CRS|2 , eq. (6.17), as black solid line for σ = 1/3 and σ = 1/2
with dots marking the values at τ1 ∼
= 681.21 and τ2 ∼
= 681.89. Here, τ2 is chosen at a zero of ζ.
2
The red dashed line indicates the exact curve MRS
|ζ|2 . In the bottom pictures KRS, eq. (6.18),
is depicted for the different combinations of σ = 1/3 and σ = 1/2 with τ1 and τ2 . The shapes
are for σ = 1/2 (two right pictures) symmetric in p and the positive areas clearly dominate for
τ1 , whereas at τ2 the positive and negative contributions cancel each other when integrated over
phase space, indicating the zero of ζ. Although the structures for σ = 1/3 (two left pictures)
are less pronounced than for σ = 1/2, the positive contributions dominate in both pictures as
tab. 6.1 confirms. For the sake of simplicity, we have chosen ν = n0 = 10.
In the bottom row of fig. 6.3, we present KRS for σ = 1/3 and σ = 1/2 with ν = n0 = 10
for two different values of τ . Since τ2 is chosen at a zero of ζ, the positive and negative
areas in the fourth picture cancel each other quite well when we integrate over phase
space. This is also confirmed by the analysis of the data in tab. 6.1. In the other pictures,
the positive areas dominate. Although the structures for σ = 1/3 are less pronounced,
the values calculated in tab. 6.1 for τ1 are nearly the same for both σ, but for τ2 the
value for σ = 1/3 is about 25-times larger. This strongly indicates that ζ(1/3 + iτ2 )
cannot be zero.
The overlap |CRS|2 , defined by eq. (6.17), is depicted as black solid line in the pictures
above with dots marking the values at τ1 and τ2 . Due to the approximate character
of the Riemann-Siegel formula, the black line differs from the red dashed line which
2 |ζ(σ + iτ )|2 .
indicates the exact curve MRS
49
6. Riemann-Siegel states
σ
τ1
τ2
1/3
R R
2π dx dp KRS
2
MRS
|ζRS|2
|ζ|2
0.151
0.00398
37.80
39.99
1/2
0.154
0.00992
15.54
16.53
1/3
0.00374
0.00398
0.9397
0.8267
1/2
0.000149
0.00992
0.01499
0
Table 6.1.: The values in the middle columns result from the data shown in the phase space
pictures of fig. 6.3. The square of the absolute value |ζRS|2 can either be calculated from |CRS|2 ,
eq. (6.17), or directly with the Riemann-Siegel formula (6.1). The integration over KRS results
for τ1 in similar values for both σ. However, the value at τ2 is for σ = 1/3 about 25-times
larger than for σ = 1/2, coinciding with the fact that the negative and positive contributions
cancel each other quite well in the phase space picture of KRS for σ = 1/2 at τ2 . Due to
the approximate character of the Riemann-Siegel formula, the integral does not vanish entirely,
causing the deviations to the exact values |ζ|2 .
In the next section, we apply the definition of the Moyal functions to get a complexvalued expression for the overlap CRS itself.
6.6. Overlap in Moyal representation
We have shown in section 3.3.1 that we can express the overlap (3.11) of two entangled
states of the form (3.5) in terms of the Moyal functions of the oscillator parts. Hence,
the complex-valued expression (6.4) is given by the phase space integration
Nn
CRS = √ 0
2
Z∞
−∞
dx
Z∞
−∞
dp K RS = MRS ζRS
(6.19)
over the kernel
K RS ≡
i
Nν (σ) h
W|σ,τ in0 hσ,0| + γ(σ, τ ) W|σ,−τ in0 h2−3σ,0|
Nn0 (σ)
(6.20)
of two Moyal functions of the form
W|σ1 ,τ in0 hσ2 ,0| = Nn0 (σ1 ) Nn0 (σ2 )
nX
0 −1
e −iτ ln(n+1)
W|nihm| .
σ1 /2 (m + 1)σ2 /2
(n
+
1)
n,m=0
(6.21)
Here, we have used the property
Z∞
−∞
dx
Z∞
−∞
dp W|σ1 ,τ iν n0 hσ2 ,0|
Nν (σ1 )
=
Nn0 (σ1 )
Z∞
−∞
dx
Z∞
dp W|σ1 ,τ in0 hσ2 ,0|
(6.22)
−∞
of the Moyal function of two truncated Riemann states with n0 ≤ ν. This equation
reflects the fact that the Fock states with ν > n0 do not contribute to the overlap
50
6.6. Overlap in Moyal representation
n0 hσ2 , 0|σ1 , τ iν .
The truncation parameter ν only appears in the normalization Nν (σ1 )
of the state |σ1 , τ iν and therefore acts as a factor.
Moreover, when we rewrite eq. (6.21) by
"n −1
0
X
e −iτ ln(n+1) W|ni
W|σ1 ,τ in0 hσ2 ,0| = Nn0 (σ1 ) Nn0 (σ2 )
(n + 1)(σ1 +σ2 )/2
n=0
#
nX
0 −1 n−1
X e −iτ ln(n+1) W|nihm| + e −iτ ln(m+1) W |nihm|
+
,
σ1 /2 (m + 1)σ2 /2
(n
+
1)
n=1 m=0
we find that the Wigner functions W|ni in the first sum are multiplied by a timedependent factor, like in the case of the Moyal function W|σ,τ iν hσ,−τ | , eq. (6.13), connected
to the off-diagonal Wigner matrix elements of the Riemann-Siegel state. In contrast to
eq. (6.13), the time-dependent factors in the double sum result here in a rotation of the
Moyal functions W|nihm| . This behavior is well-known from the rotation of the Wigner
function W|σ,τ iν , eq. (5.6), but now the rotation angle is ϕn0 = τ ln(n + 1), eq. (4.14),
which depends only on one summation index. Since W|nihm| and its complex conjugated
are rotated by different angles, W|σ1 ,τ in0 hσ2 ,0| has in general no symmetries.
We show in fig. 6.4 the real and imaginary part of the two contributions W|σ,τ in0 hσ,0|
and W|σ,−τ in0 h2−3σ,0| of K RS, eq. (6.20), for σ = 1/3 and n0 = 10 at four different
times. Besides the times τ1 and τ2 , which are the same as in fig. 6.3, we depict also the
Moyal functions at τ = 680 and τ = 683 to give a flavor of the rotation. As expected,
W|1/3,τ i10 h1/3,0| rotates clockwise whereas W|1/3,−τ i10 h1,0| rotates counter-clockwise due to
the different signs of τ .
With the help of eq. (6.9) and eq. (6.21), the kernel (6.20) reads
"
#
nX
0 −1
W|nihm|
e −iτ ln(n+1) χ(σ + iτ ) e iτ ln(n+1)
K RS = Nν (σ) Nn0 (σ)
+
(6.23)
σ/2
(2−3σ)/2
(m + 1)
(m + 1)
(n + 1)σ/2
n,m=0
in terms of Fock states. Although each contributing Wigner function is in general not
symmetric in p, we find for σ = 1/2 the connection
K RS (x, −p) = e i arg{χ(1/2+iτ )} K RS (x, p)
(6.24)
since W|nihm| (x, −p) = W|nihm| (x, p), eq. (4.13). Hence, the absolute value of K RS is on
the critical line symmetric in p, as fig. 6.5 confirms, where we depict K RS by its real and
imaginary part (upper rows) as well as its absolute value and phase (bottom rows) for
the same values of σ and τ as in fig. 6.3. The dominant positive or negative areas are
produced by the factor γ, eq. (6.9), in front of the Moyal function W|σ,0in0 h2−3σ| . Its real
part is at τ1 for both values of σ negative while the imaginary part almost vanishes. For
τ2 , the real part is positive and the imaginary part negative and about 10-times smaller
than the real one.
Finally, we get from eq. (6.19) the representation
1
ζRS = 2
Nn0 (σ)
Z∞
−∞
dx
Z∞
−∞
h
i
dp W|σ,τ in0 hσ,0| + γ(σ, τ ) W|σ,−τ in0 h2−3σ,0|
(6.25)
51
6. Riemann-Siegel states
Figure 6.4.: Real and imaginary part of the Moyal functions W|σ,τ in0 hσ,0| and W|σ,−τ in0 h2−3σ,0| ,
eq. (6.21), for σ = 1/3, n0 = 10 and different values of τ . Here, τ1 and τ2 are the same as
in fig. 6.3. As expected, W|σ,τ in0 hσ,0| (first and third row) rotates clockwise whereas the other
Moyal function rotates counter-clockwise.
52
6.6. Overlap in Moyal representation
Figure 6.5.: The kernel K RS of the Moyal representation, eq. (6.23), is depicted by its real and
imaginary part (top rows) as well as absolute value and argument (bottom rows) for ν = n0 = 10
and the same values of σ and τ used in fig. 6.3. In general, the structures of K RS have no
symmetries, but the absolute value is for σ = 1/2 symmetric in p, as eq. (6.24) confirms. The
structures here have more dominant positive or negative areas than the corresponding Moyal
functions in the pictures of fig. 6.4, since W|σ,−τ in0 h2−3σ,0| is for τ1 additionally multiplied with
the (almost purely) negative factors γ(1/3, τ1 ) ∼
= −1.5 and γ(1/2, τ1 ) ∼
= −1 and for τ2 by the
∼
complex values γ(1/3, τ2 ) = 1.5 − 0.13 i and γ(1/2, τ2 ) ∼
= 1 − 0.09 i. Moreover, K RS is connected
to the Riemann-Siegel formula via eq. (6.25). The explicit values calculated from the data are
given in tab. 6.2.
53
6. Riemann-Siegel states
σ
τ1
1/3
R
R
dx dp K RS
−0.2568 − 0.9548 i
Nn−2
(σ)
0
6.220
0.9887 e −1.8335 i
1/2
1/3
−8 · 10−6 − 0.7854 i
5.021
0.0707 + 0.1388 i
6.220
0.1558
1/2
0.0242
e −0.04321 i
−1.597 − 5.939 i
−1.597 − 6.119 i
−4 · 10−5 − 3.944 i
−4 · 10−5 − 4.066 i
0.4398 + 0.8631 i
0.2607 + 0.8710 i
3.944 e −1.5708 i
e 1.0995 i
0.0241 − 0.00104 i
ζ
6.150 e −1.8335 i
0.7854 e −1.5708 i
τ2
ζRS
0.9687
5.021
e 1.0995 i
0.1212 − 0.00524 i
0.1214
e −0.04321 i
6.324 e −1.8261 i
4.066e −1.5708 i
0.9092 e 1.2800 i
0
0
Table 6.2.: The values in the middle columns are calculated from the data displayed in the phase
space pictures of fig. 6.5. The explicit connection between ζRS and the integral over K RS is given
by eq. (6.25). Due to the approximate character of ζRS, the calculated values deviate from the
exact ones given in the last column.
for the Riemann-Siegel formula in terms of the Moyal functions. The dependence on the
arbitrary chosen summation limit ν vanishes completely. Thus, the limit ν = n0 for the
Riemann-Siegel state suffices to produce ζ.
In tab. 6.2, we present the values of ζRS calculated by the data shown in the phase
space pictures of fig. 6.5. Additionally, we depict the behavior of the real and imaginary
part as well as the absolute value and phase of ζRS in fig. 6.6. The blue dashed curves for
σ = 1/3 exceed the curves of σ = 1/2 given in red. Moreover, the phase is discontinuous
at the zeros of ζ on the critical line, confirmed by the Argand diagram [9] shown in
fig. 6.7. In both pictures, we have marked the values calculated in tab. 6.2 by dots.
6.7. Summary
We have shown in this chapter that the Riemann-Siegel formula (6.1) can be expressed
by the scalar product (6.4) of the Riemann-Siegel state |ΨRSi, eq. (6.3), and its reference state |ΦRSi, eq. (6.7). Due to time evolution, the Riemann-Siegel state becomes
entangled while the reference state is entangled for all σ 6= 1/2. Hence, only σ = 1/2 produces p-symmetric phase space pictures for the kernel KRS of the Wigner representation,
eq. (6.18), and quasi-symmetric structures for the Moyal representation K RS, eq. (6.24).
The physical picture therefore emphasizes the importance of the critical line.
To enhance the accuracy of the calculated values, we discuss the Berry-Keating formula, which already contains the first correction term to ζRS, in the next chapter.
54
6.7. Summary
Im {ζRS}
Re {ζRS}
8
6
4
2
680
682
680
684
682
τ
684
-2
-4
-6
|ζRS|
arg{ζRS}
Π
8
Π
2
6
4
680
2
-
680
682
τ
684
682
684
τ
Π
2
-Π
Figure 6.6.: Dependence of ζRS, eq. (6.1), on τ shown by its real and imaginary part as well as
absolute value and argument for σ = 1/3 (blue dashed line) and σ = 1/2 (red solid line). The
argument is discontinuous at the zeros of ζ, as the Argand diagram in fig. 6.7 shows. The values
marked by dots are the ones calculated in tab. 6.2.
Im ζ
6
4
2
-2
2
4
6
8
Re ζ
-2
-4
-6
Figure 6.7.: The Argand diagram of ζ confirms that there is no zero for σ = 1/3 (blue dashed
line) but one for σ = 1/2 (red curve) in the interval 680 ≤ τ ≤ 683.5. The curves start at values
with positive imaginary part and proceed in clockwise direction, indicated by the arrows. The
dots mark the values calculated in tab. 6.2.
55
7. Berry-Keating reference states
In section 1.6, we have shown that the approximation of the Riemann zeta function can
be improved by using the Berry-Keating formula ζBK. Since it is similar to the RiemannSiegel formula, we can produce ζBK from the overlap of the Riemann-Siegel state with
the Berry-Keating reference state, which we derive in this chapter. The analysis of the
corresponding Wigner and Moyal representations confirm the enhanced accuracy of the
approximation.
7.1. Definition
With definition (1.7) of the symmetric representation, the Berry-Keating representation
of Z, eq. (1.12), yields
"
#
nX
B −1
e −iτ ln(n+1)
e iτ ln(n+1)
ζBK(σ + iτ ) =
E(n, τ ) + χ(σ + iτ )
E (n, τ )
(7.1)
(n + 1)σ
(n + 1)1−σ
n=0
the Berry-Keating formula of the Riemann zeta function. The abbreviation E(n, τ ),
given by eq. (1.13), contains the complementary error function [50] while the summation
limit fulfills nB ≥ n0 . For the sake of simplicity, we choose nB = n0 (τ ) whenever we
compare the Berry-Keating representations to the Riemann-Siegel ones.
Since eq. (7.1) is reminiscent of the Riemann-Siegel formula (6.2), we use the RiemannSiegel state |ΨRS(σ, τ )i, eq. (6.3), to calculate the overlap
CBK ≡ hΦBK(σ, τ )|ΨRS(σ, τ )i = MBK ζBK(σ + iτ ) .
Comparison of both sides leads to the definition
en (σ, τ ) |φ (σ, τ )in |ei + γ (σ, τ ) |φ (2 − 3σ, τ )in |gi
|ΦBK(σ, τ )i ≡ N
B
B
B
B
B
B
(7.2)
(7.3)
of the Berry-Keating reference state, where we have introduced the Berry states
|φB(σ, τ )inB
en (σ, τ )
≡N
B
nX
B −1
n=0
E (n, τ )
|ni
(n + 1)σ/2
(7.4)
with n0 ≤ nB ≤ ν and the normalization
en (σ, τ ) ≡
N
B
"n −1
#−1/2
B
X
|E(n, τ )|2
n=0
(n + 1)σ
.
(7.5)
57
7. Berry-Keating reference states
The Berry states |φBi are given by eq. (7.4) with E replaced by its complex conjugate E.
In analogy to the normalization of the Riemann-Siegel reference state, eq. (6.8), the
normalization of the Berry-Keating reference state reads
but now the factor
1
en (σ, τ ) ≡ q
,
N
B
2
1 + |γB(σ, τ )|
γB(σ, τ ) ≡
en (σ, τ )
N
B
en (2 − 3σ, τ )
N
B
χ(σ + iτ )
(7.6)
(7.7)
en of the Berry states, eq. (7.5). As
is involved, which depends on the normalizations N
B
in the case of the Riemann-Siegel reference state, we have to interpret τ in the definition
(7.3) of the reference state |ΦBKi as the imaginary part of s and not as time.
Finally, we identify the proportionality factor
MBK ≡
Nν (σ) e
en (σ, τ ) .
√
NnB (σ, τ ) N
B
2
In contrast to the last chapter where the Riemann-Siegel reference state consists of the
truncated Riemann state |σ, 0in0 and |2−3σ, 0in0 defined by eq. (5.4), the Berry-Keating
reference state |ΦBKi depends on the Berry states |φB(σ, τ )inB and |φB(2−3σ, τ )inB which
contain additionally the n-dependent factors E(n, τ ) and E(n, τ ) in their probability
amplitudes. Due to these factors, the Berry-Keating reference state is always entangled
while the Riemann-Siegel reference state, eq. (6.7), becomes separable on the critical
line, as we show in the appendices E.4 and E.5.
Since we already gave a detailed analysis of the Wigner matrix Ŵ|ΨRSi of the RiemannSiegel state in section 6.3, we restrict ourselves to the investigation of the Wigner representations of the Berry states and the Berry-Keating reference state in the next sections.
Afterwards, we describe the overlap in Wigner and Moyal representation which produces
the Berry-Keating formula (7.1).
7.2. Wigner function of Berry state
According to eq. (3.14), the Wigner functions of the Berry states, eq. (7.4), are given by
"n −1
B
X
|E(n, τ )|2 W|ni
2
e
W|φB(σ,τ )inB ≡ NnB (σ, τ )
(n + 1)σ
n=0
#
nX
B −1 n−1
X Re E (n, τ ) E(m, τ ) W|nihm|
+2
.
(7.8)
[(n + 1)(m + 1)]σ/2
n=1 m=0
Since the values of E are equal to one for almost all n < n0 , as shown in appendix F.2,
the shapes are reminiscent of the ones for the Wigner functions of the initial truncated
Riemann state |σ, 0inB defined by eq. (5.6). Nevertheless, the τ -dependence of E creates
58
7.2. Wigner function of Berry state
Figure 7.1.: Difference W|φB(σ,τ )in0 −W|σ,0in0 between the Wigner functions, eqs. (7.8) and (5.6),
of the Berry state |φBin0 and the initial truncated Riemann state |σ, 0in0 for σ = 1/2 and
different τ determining the summation limit n0 = n0 (τ ). At the jumps 2πn20 of the RiemannSiegel formula, we find the largest differences between the two states. They become smaller
in the interval 2π n20 ≤ τ < 2π(n0 + 1)2 and rotate due to the τ -dependent factors E(n, τ ) in
W|φB(σ,τ )in0 . Moreover, the deviations fade for increasing summation limit.
small deviations which make W|φBinB asymmetric in p. To emphasize the inequality of the
two states, we depict the difference W|φB(1/2,τ )in0 −W|1/2,0in0 between the Wigner function
of the Berry state |φBin0 and the initial truncated Riemann state |σ, 0in0 , eqs. (7.8) and
(5.6), in fig. 7.1 for different values τ and summation limits n0 (τ ) up to four. The
pictures show that the deviations are distinct at the jumps τ = 2π n20 of the RiemannSiegel formula and fade in the interval 2πn20 ≤ τ < 2π(n0 + 1)2 before the next jump.
Of course, increasing summation limits result in more complicated, but less pronounced
structures.
The same properties hold for the Berry state |φBinB . Indeed, when we recall that
W|nihm| (x, −p) = W|nihm| (x, p), a closer look at the Wigner function
W|φ
(σ,τ )inB
B
≡
en2 (σ, τ )
N
B
"n −1
B
X
|E(n, τ )|2 W|ni
(n + 1)σ
n=0
#
nX
B −1 n−1
X Re E(n, τ ) E (m, τ ) W|nihm|
+2
[(n + 1)(m + 1)]σ/2
n=1 m=0
reveals that it is the opposite hand of W|φB(σ,τ )inB with respect to the x-axis, that is
W|φ
(σ,τ )inB (x, p)
B
= W|φB(σ,τ )inB (x, −p) .
(7.9)
This relation makes the Wigner matrix of |ΦBKi special on the critical line, as we will
see in the next section.
59
7. Berry-Keating reference states
7.3. Wigner matrix
Equation (3.12) yields the Wigner matrix
2
e
Ŵ|ΦBKi ≡ NnB W|φB(σ,τ )inB |eihe| + |γB|2 W|φ
(2−3σ,τ )inB
B
+ γB W|φ
(σ,τ )inB hφB(2−3σ,τ )| |eihg|
B
+ γ B W|φ
|gihg|
(σ,τ )| |gihe|
(7.10)
(2−3σ,τ )inB hφB
B
for the Berry-Keating reference state |ΦBKi, eq. (7.3).
The diagonal matrix element connected to the excited state |ei is proportional to the
Wigner function of the Berry state |φB(σ, τ )inB , eq. (7.8), whereas the one connected to
the ground state |gi is the opposite hand of W|φB(2−3σ,τ )inB , due to eq. (7.9). Hence,
on the critical line, the diagonal matrix elements are the mirror images√of each other,
en = 1/ 2, eq. (7.6), is
since for σ = 1/2 eq. (7.7) simplifies to γB = 1 and, therefore, N
B
independent of τ .
In fig. 7.2, we depict the Wigner matrix elements of the Berry-Keating reference state
|ΦBKi for σ = 1/3, τ = τ1 ∼
= 681.21 and nB = 10 on the left. As expected, the shapes
resemble the ones of the Riemann-Siegel reference state shown in fig. 6.2, but the pictures
on the right reveal the small deviations calculated from the difference
Ŵ|ΦBKi − Ŵ|ΦRSi ≡ W|ei |eihe| + W|gi |gihg| + W|eihg| |eihg| + W|gihe| |gihe|
(7.11)
of the Wigner matrices, eqs. (7.10) and (6.15), with nB = n0 (τ ).
The two bottom rows present the real and imaginary part of the off-diagonal matrix element connected to |eihg|. It is proportional to the Moyal function W|φ (σ,τ )in hφ (2−3σ,τ )|
B
B
B
defined by
W|φ
(σ ,τ )inB hφB(σ2 ,τ )|
B 1
en (σ1 , τ ) N
en (σ2 , τ )
≡ N
B
B
nX
B −1
n,m=0
E (n, τ ) E (m, τ ) W|nihm|
(n + 1)σ1 /2 (m + 1)σ2 /2
.
(7.12)
This expression is only symmetric in p on the critical line, just like the Moyal function
W|σ,0in0 h2−3σ,0| , eq. (6.16), in the case of the Riemann-Siegel reference state. For σ 6= 1/2,
asymmetries appear, caused by the factors E in the Moyal function and additionally by
the factor γB containing the function χ. Thus, the shapes of the off-diagonal matrix
element in fig. 7.2 again only seem to be antisymmetric.
7.4. Overlap in Wigner representation
Finally, the Wigner representations Ŵ|ΨRSi and Ŵ|ΦBKi , eqs. (6.11) and (7.10), of the
Riemann-Siegel state and the Berry-Keating reference state yield the absolute value of
the overlap (7.2)
Z∞ Z∞
2
|CBK| = 2π~ dx dp KBK = MBK
|ζBK(σ + iτ )|2 ,
2
−∞
60
−∞
(7.13)
7.4. Overlap in Wigner representation
Figure 7.2.: The Wigner matrix elements of the Berry-Keating reference state |ΦBKi are shown
on the left for σ = 1/3, τ = τ1 ∼
= 681.21 and nB = 10. To avoid cumbersome notation, we have
used the abbreviations |φBi ≡ |φB(σ, τ )inB and |φBi ≡ |φB(2 − 3σ, τ )inB . Although the shapes
all look like the ones of the Riemann-Siegel reference state |ΦRSi for the same values (fig. 6.2),
there is a small difference defined by eq. (7.11) and depicted on the right. The differences of
the diagonal matrix elements (two upper right pictures) emphasize that the ones of the BerryKeating reference state are not symmetric in p, in contrast to the ones of the Riemann-Siegel
state. Moreover, the shapes of the off-diagonal matrix element (two bottom rows) are only nearly
antisymmetric, since the τ -dependence of E in eq. (7.12) and the multiplication with γB makes
them asymmetric with respect to p.
61
7. Berry-Keating reference states
1/3
R R
2π dx dp KBK
0.158
2
MBK
0.00401
|ζBK|2
39.31
37.80
39.99
1/2
0.163
0.01003
16.22
15.54
16.53
1/3
0.003
0.00401
0.8552
0.9397
0.8267
1/2
1.5 · 10−5
0.01003
0.00152
0.01499
0
σ
τ1
τ2
|ζRS|2
|ζ|2
Table 7.1.: The values in the middle columns result from the data shown in the phase space
pictures of fig. 7.3. For |ζBK|2 they can either be calculated from eq. (7.13) or directly with the
Berry-Keating formula (7.1). The values confirm the improvement of the Berry-Keating formula
compared to the Riemann-Siegel formula, eq. (6.1).
with the help of eq. (3.15). Here, the kernel reads
KBK ≡
e2 h
N
nB
W|σ,τ iν W|φB(σ,τ )inB + |γB|2 W|σ,−τ iν W|φ (2−3σ,τ )in
B
B
2
n
oi
+ 2 Re γB W|σ,τ iν hσ,−τ | W|φ
(σ,τ )inB hφB(2−3σ,τ )|
B
(7.14)
according to eq. (3.16). Like the kernel KRS, eq. (6.18), producing the Riemann-Siegel
formula, the phase space pictures of KBK are symmetric in p only if σ = 1/2, due to the
symmetries of the Wigner and Moyal functions given by eqs. (6.12) and (7.9) as well as
eqs. (6.14) and (7.12), respectively.
Figure 7.3 shows KBK for nB = 10 and the combinations of σ = 1/3 and σ = 1/2 with
τ1 ∼
= 681.21 and τ2 ∼
= 681.89 in the second row. The pictures are more pronounced for
σ = 1/2 and, especially for τ2 , the shapes differ from the ones of KRS which are repeated
in the third row. The bottom pictures present the quite large difference KBK − KRS for
σ = 1/2. For σ = 1/3, the values only deviate by |KBK − KRS| < 0.0006.
Finally, the first row displays the absolute value of the overlap |CBK|2 , eq. (7.13), in
blue with dots marking the values we get from integration over the phase space pictures
2 |ζ|2 is plotted as red dashed line and M2 |ζ |2
of KBK. Moreover, the exact curve MBK
RS
BK
as black dotted line. As expected, we get more accurate values from the Berry-Keating
formula than from the Riemann-Siegel approximation. This is also confirmed by tab. 7.1
which shows the values calculated from the phase space pictures in comparison to the
ones evaluated by the formulae.
7.5. Overlap in Moyal representation
The definition (3.11) of the overlap hΦBK|ΨRSi leads to the Moyal representation
∞
Z∞
en Z
N
B
CBK = √
dx
dp K BK = MBK ζBK
2
−∞
62
−∞
7.5. Overlap in Moyal representation
Figure 7.3.: We depict in the second row the kernel KBK, eq. (7.14), for nB = 10 and the same
values of σ and τ as used in fig. 6.3 for the kernel KRS (pictures repeated in the third row). In
contrast to KRS, the pictures of KBK for σ = 1/2 are less pronounced. Moreover, the distribution
of the positive and negative regions in phase space is quite different for τ2 . The bottom pictures
show the difference KBK − KRS for σ = 1/2. For σ = 1/3, we get only very small differences with
|KBK − KRS| < 0.0006. However, integration of KBK over phase space according to eq. (7.13) yields
2
more accurate values compared to the exact ones MBK
|ζ|2 , as tab. 7.1 confirms. In the upper
2
2
pictures, the red dashed curves indicate MBK |ζ| while the blue curves represent |CBK|2 with dots
marking the values for τ1 and τ2 . To emphasize the improvement of the approximation by the
2
Berry-Keating formula, we indicate MBK
|ζRS|2 as black dotted line.
63
7. Berry-Keating reference states
of the Berry-Keating formula. Here, the kernel of the phase space integral is given by
i
Nν (σ) h
K BK ≡
W|σ,τ inB hφB(σ,τ )| + γB(σ, τ ) W|σ,−τ in hφ (2−3σ,τ )|
(7.15)
B
B
NnB (σ)
with the Moyal functions
en (σ2 , τ2 )
W|σ1 ,τ1 inB hφB(σ2 ,τ2 )| = NnB (σ1 ) N
B
The corresponding Moyal function W|σ1 ,τ1 in
nX
B −1
e −iτ1 ln(n+1) E(m, τ2 )
W|nihm| .
σ1 /2 (m + 1)σ2 /2
(n
+
1)
n,m=0
B
hφB(σ2 ,τ2 )|
(7.16)
emerges from eq. (7.16) by sub-
stituting E with E .
In definition (7.15) of the kernel, the prefactor Nν /NnB results form the property
Z∞
−∞
dx
Z∞
−∞
dp W|σ1 ,τ1 iν nB hφB(σ2 ,τ2 )|
Nν (σ1 )
=
NnB (σ1 )
Z∞
dx
−∞
Z∞
dp W|σ1 ,τ1 inB hφB(σ2 ,τ2 )|
−∞
for nB ≤ ν which also holds if we replace |φBinB by |φBinB . Analogous to eq. (6.22), this
expression reflects the fact that the Fock states with ν > nB do not influence the overlap
nB hφB(σ2 , τ2 )|σ1 , τ1 iν .
The first and second row of fig. 7.4 depict the real and imaginary part of the
Moyal functions W|σ,τ inB hφB(σ,τ )| and W|σ,−τ in hφ (2−3σ,τ )| appearing in eq. (7.15) of
B
B
the kernel K BK for σ = 1/3, τ = τ1 and nB = 10. The pictures below reveal the tiny
difference between these Moyal functions and the Moyal functions W|1/3,τ1 i10 h1/3,0| and
W|1/3,−τ1 i10 h1,0| of the truncated Riemann states, defined by eq. (6.21) and shown in
fig. 6.4. The additional factor E in the Moyal functions containing the Berry states,
eq. (7.16), result in a larger rotation of some of the Moyal functions W|nihm| . The
direction of rotation depends, as in the Riemann-Siegel case, on the sign of τ in the
truncated Riemann state |σ, τ iν .
Finally, we express the kernel, eq. (7.15), in terms of Fock states and arrive at
"
#
nX
0 −1
−iτ ln(n+1) E(m, τ )
iτ ln(n+1) E (m, τ )
W|nihm|
e
χ(σ
+
iτ
)
e
en
K BK = Nν N
+
.
B
σ/2
(2−3σ)/2
σ/2
(m
+
1)
(m
+
1)
(n
+
1)
n,m=0
On the critical line, this yields the relation
K BK (x, −p) = e i arg{χ(1/2+iτ )} K BK (x, p)
analogous to eq. (6.24). Hence, the absolute value of K BK is symmetric in p.
In fig. 7.5, we show only the real and imaginary part of K BK and its connection
1
ζBK =
e
Nν (σ) NnB (σ, τ )
64
Z∞
−∞
dx
Z∞
−∞
dp K BK
(7.17)
7.5. Overlap in Moyal representation
Figure 7.4.: The real and imaginary part of the Moyal functions W|σ,τ inB hφB(σ,τ )| and
W|σ,−τ in hφ (2−3σ,τ )| are depicted in the two top rows for σ = 1/3, τ = τ1 and nB = 10. In
B
B
the bottom rows, we present the differences ∆WφB ≡ W|1/3,τ1 i10 hφ(1/3,τ1 )| − W|1/3,τ1 i10 h1/3,0| and
∆Wφ ≡ W|1/3,−τ1 i10 hφ (1,τ1 )| − W|1/3,−τ1 i10 h1,0| between the Moyal functions shown above and
B
the Moyal functions displayed in fig. 6.4. The differences are quite small, but they illustrate that
the factor E in eq. (7.16) results in an additional rotation of some contributing W|nihm| .
65
7. Berry-Keating reference states
Figure 7.5.: Here and on the next page, we show the real and imaginary part of ζ BK for σ = 1/3
and σ = 1/2 in the first row, with dots marking the values at τ1 ∼
= 681.21 and τ2 ∼
= 681.89. In
the insets, the differences ∆ζ of ζBK − ζ and ζRS − ζ are depicted as blue and as black dotted line,
respectively. The red dashed vertical lines indicate τ1 and τ2 . The row below presents the real or
66
7.5. Overlap in Moyal representation
imaginary part of K BK, eq. (7.15), as phase space pictures which are connected to ζBK via
eq. (7.17). They are reminiscent of the ones for K RS, fig. 6.5, of the Riemann-Siegel representation, repeated in the third row. However, the kernels are not the same as the bottom row
confirms, where the difference K BK − K RS is depicted.
67
7. Berry-Keating reference states
σ
τ1
1/3
1/2
τ2
1/3
R
R
dx dp K BK
−0.2584 − 0.9813 i
−8 ·
10−6
− 0.807 i
0.0513 + 0.1405 i
en−1
Nn−1
N
B
B
6.180
4.994
6.181
ζBK
ζ and (ζRS)
−1.597 − 6.064 i
−1.597 − 6.119 i
−4 ·
10−5
− 4.029 i
0.3182 + 0.8683 i
(−1.597 − 5.939 i)
−4 · 10−5 − 4.066 i
(−4 · 10−5 − 3.944 i)
0.2607 + 0.8710 i
(0.4398 + 0.8631 i)
1/2
0.0076 − 0.0003 i
4.995
0.0382 − 0.0017 i
0
(0.1212 − 0.00524 i)
Table 7.2.: The values in the middle columns are calculated from the data displayed in the phase
space pictures of fig. 7.5 and connected to ζBK via eq. (7.17). The right column shows that the
values ζBK are much closer to the exact values of ζ than the values of ζRS given in brackets.
to the Berry-Keating representation ζBK, eq. (7.1). The corresponding values are displayed in tab. 7.2. To emphasize the difference of the shapes to the ones depicted in
fig. 6.5 for the Riemann-Siegel representation, we repeat K RS in the third row and present
the difference K BK − K RS beneath. Although the difference of the kernels is small compared to the phase space pictures above, the insets in the first row emphasizes that the
deviation ζBK − ζ (blue curve) is smaller than ζRS − ζ (black dotted line), confirming the
improvement of the approximation by the Berry-Keating formula.
7.6. Summary
Taking into account that the function E effectively cuts the sums at n0 and E(n, τ ) = 1
for almost all n < n0 , the modifications of the Berry-Keating formula in contrast to the
Riemann-Siegel formula are small. However, these little changes enhance the accuracy
of the approximation by including the first order correction term of the Riemann-Siegel
approximation. The price we have to pay for this improvement in the physical picture is
that the Berry-Keating reference state is always entangled. Nevertheless, we can identify
the critical line: the Wigner matrix elements as well as the absolute value of the kernel
K BK in the Moyal representation are only symmetric in p for σ = 1/2.
We conclude the analysis of zeta states by mentioning that a further enhancement
of the accuracy in the physical description can be achieved by inclusion of higher order
correction terms. The investigation of the resulting states is left for future work.
In the next chapter, we suggest a modification of the Jaynes-Cummings-Paul model to
create the Riemann Hamiltonian which is essential in the realization of the zeta states.
68
8. Jaynes-Cummings-Paul approach to the
Riemann Hamiltonian
The eye-catching similarity between the phase of the Riemann zeta function and the
time evolution produced by the effective Hamiltonian of the Jaynes-Cummings-Paul
(JCP) model suggests that modification of this outstanding approach should indicate
the physical realization of the Riemann Hamiltonian ĤR . Hence, we give in this chapter
a short review of the JCP model and then show how we have to change the interaction
to generate the Riemann Hamiltonian.
8.1. Jaynes-Cummings-Paul model
In 1963, Edwin T. Jaynes together with Fred W. Cummings [16] and, at the same time
independently, Harry Paul [17] introduced a model describing the interaction of an atom
with a quantized light field. Although it is the most elementary model for the interaction
– it considers only a fixed two-level atom and a single mode of the radiation field – the
JCP model is able to describe most phenomena in cavity quantum electrodynamics.
Thus, ‘it has become the drosophila of quantum optics.’ [28]
The JCP model makes use of some simplifications [45]: (i) The internal structure of
the atom is depicted by only two energy levels, the excited state |ei and the ground
state |gi, which are separated by an energy gap of ~ωa , see fig. 8.1. Hence, the free
Hamiltonian1 of the atom is given by
Ĥa ≡
~ωa
σ̂z ,
2
(8.1)
where σ̂z , eq. (3.3), is the Pauli spin operator.
(ii) We consider only a single mode of the radiation field
~ˆ ≡ E~0 (â + ↠)
E
(8.2)
with E~0 containing the vacuum electric field and the mode function of the cavity C. This
yields the free Hamiltonian of the field
Ĥfield ≡ ~Ω â† â .
1
(8.3)
There is, of course, an additional contribution in Ĥa which is proportional to the identity operator
1̂. This contribution yields overall phases which can be neglected since they cancel themselves in
the interaction picture which we use to determine the effective Hamiltonian. We neglect the vacuum
energy of the cavity field for the same reason.
69
8. Jaynes-Cummings-Paul approach to the Riemann Hamiltonian
~ωa
ei
~ωa
gi
|ei
|gi
~
E
C
Figure 8.1.: The two-level atom in the cavity C has an energy gap of ~ωa between the excited
state |ei and the ground state |gi.
(iii) The interaction between atom and field is approximated by the dipole interaction
~ˆ
Ĥint ≡ −e~rˆ E
(8.4)
of the two level atom with the quantized field, eq. (8.2), in the cavity C in which (iv)
the motion of the atom is neglected. In terms of the Pauli operator σ̂ ≡ |gihe| the dipole
moment is
e~rˆ = p~ σ̂ † + p~ ∗ σ̂
and eq. (8.4) then reads
~
Ĥint = ~g σ̂ + σ̂ † â + â†
with the constant g = |~p ~E0 | coupling the dipole moment to the electric field. Both vectors
are chosen to be anti-parallel to achieve the sum of the Pauli operators as well as the
sum of the annihilation and creation operator â and ↠, respectively.
A transformation of the interaction Hamiltonian into the interaction picture via the
free Hamiltonian Ĥ0 = Ĥa + Ĥfield , eqs. (8.1) and (8.3), leads to
i
i
I
Ĥint
≡ e ~ Ĥ0 t Ĥint e − ~ Ĥ0 t = ĤJC + ĤaJC .
(8.5)
The first part, the so-called Jaynes-Cummings Hamiltonian
ĤJC = ~g ↠σ̂ e −i(ωa −Ω)t + h.c. ,
consists of the energy conserving terms whereas the second contribution, the anti-JaynesCummings Hamiltonian
ĤaJC = ~g â σ̂ e −i(ωa +Ω)t + h.c. ,
annihilates a photon while transferring the atom into the ground state. This ‘unphysical’
behavior of the second term is often neglected based on the rotating wave approximation
[28]: Due to the larger frequency ωa + Ω, the anti-Jaynes-Cummings part rotates faster
70
8.2. Riemann Hamiltonian
than the Jaynes-Cummings contribution with frequency |ωa − Ω|, leading to negligible
contributions.
We mention here that Zheng-Hong Li et al. [51] propose in their paper a transformation
of the interaction Hamiltonian quite similar to the transformation into the interaction
picture with the advantage that it erases one of the contributions completely without
using approximations. However, we will not follow this approach, but use second order
perturbation theory to derive the effective Hamiltonian.
I , eq. (8.5), is of the form
The interaction Hamiltonian Ĥint
X
I
Ĥint
(t) ≡
Ĥj σ̂ e −iνj t + Ĥj† σ̂ † e iνj t
(8.6)
j
which, in the far off-resonant case, produces the effective Hamiltonian
i
X 1 h †
Ĥeff ≡
Ĥj Ĥj − Ĥj Ĥj† 1̂ + Ĥj† Ĥj + Ĥj Ĥj† σ̂z
2~νj
(8.7)
j
as we show in appendix G.2. Hence, we get from eq. (8.5)
1
~g̃˜
ĤJC,eff = ~g̃ n̂ +
σ̂z +
1̂ ,
2
2
h
i
h
i
where g̃ ≡ g 2 (ωa − Ω)−1 + (ωa + Ω)−1 and g̃˜ ≡ g 2 (ωa − Ω)−1 − (ωa + Ω)−1 . When
we now use the rotating wave approximation to neglect the terms with (ωa + Ω)−1 we
arrive at the well-known result [28]. Since constant terms produce overall phases,
ĤJC,eff =
~ g2
n̂ σ̂z
ωa − Ω
is frequently referred to as effective Jaynes-Cummings Hamiltonian.
In the next section, we derive in the same way an effective Hamiltonian which reflects,
at least approximately, the logarithmic behavior of the Riemann Hamiltonian.
8.2. Riemann Hamiltonian
To get the Riemann Hamiltonian ĤR , eq. (3.2), as effective Hamiltonian of a slightly
changed Jaynes-Cummings-Paul model, we start again in the Schrödinger picture, but
now with the Hamiltonian
ĤS ≡ Ĥ0 + fˆ · σ̂ + σ̂ † ,
(8.8)
where once more Ĥ0 ≡ ~Ω n̂ + ~ω2 a σ̂z . The Hermitian operator fˆ only acts in the Hilbert
space of the field. Transformation into the interaction picture yields
I
Ĥint
=
∞
X
hm|fˆ|ni e −iΩt(n−m) |mihn|
n,m=0
σ̂ e −iωa t + σ̂ † e iωa t
.
71
8. Jaynes-Cummings-Paul approach to the Riemann Hamiltonian
Here, we have used the completeness relation of the Fock states to transform fˆ.
Substituting j = n − m, we find that the interaction Hamiltonian
I
Ĥint
=
∞
X
Ĥ−j σ̂ e
−iν−j t
j=0
+
∞
X
Ĥj σ̂ e −iνj t + h.c.
j=1
is of the form (8.6) with frequencies νj ≡ ωa + jΩ and the approximations
Ĥj ≡
∞
X
n=j
hn − j|fˆ|ni |n − jihn|
Ĥ−j ≡ Ĥj† .
as well as
(8.9)
Hence, the effective Hamiltonian, eq. (8.7), simplifies to


∞
∞
X
X
Ĥeff = 
εj(+) Ĥj† Ĥj + Ĥj Ĥj† + ε0 Ĥ02  σ̂z +
ε(j−) Ĥj† Ĥj − Ĥj Ĥj† 1̂ ,
j=1
j=1
where we have introduced the inverse energies
1
1
1
(±)
±
and
εj ≡
2~ νj
ν−j
ε0 ≡
1
.
~ν0
With eq. (8.9), we arrive at


∞ X
∞
∞ 2 2
X
X
ˆ Ĥeff =
ε(+) hn − j|fˆ|ni |nihn| + |n − jihn − j| + ε0
hn|f |ni |nihn| σ̂z
j
n=0
j=1 n=j
∞ X
∞
2 X
(−) ˆ
+
εj hn − j|f |ni |nihn| − |n − jihn − j| 1̂ .
j=1 n=j
Before we can compare this result to the Riemann Hamiltonian, eq. (3.2), we have to
substitute n − j in the bras and kets |n − jihn − j| by n and then interchange the
summation over j and n. This yields
∞ X
Ĥeff =
Fn σ̂z + Cn 1̂ |nihn|
(8.10)
n=0
with the abbreviations
n
∞
2
2 X
ˆ 2 X (+) (+) ˆ
ˆ
Fn ≡ ε0 hn|f |ni +
εj hn − j|f |ni +
εj hn|f |n + ji
j=1
(8.11)
j=1
and
Cn ≡
72
n
X
j=1
∞
2 X
2
ε(j−) hn − j|fˆ|ni −
ε(j−) hn|fˆ|n + ji .
j=1
(8.12)
8.2. Riemann Hamiltonian
So far, the only approximation we have used is ν −2 ν −1 to get the form (8.7) of the
effective Hamiltonian. When we now compare eq. (8.10) to the Riemann Hamiltonian
ĤR = ~ω
∞
X
ln(n + 1)|nihn| σ̂z ,
n=0
we find that the coefficients of σ̂z must fulfill the equation
!
Fn = ~ω ln(n + 1) ≡ F (n)
(8.13)
in which the frequency ω can be chosen freely. Moreover, the coefficients Cn , eq. (8.12),
must be constant or negligible.
We will answer the question how to choose fˆ in what follows.
8.2.1. Exact solution
The simplest way to solve eq. (8.13) is to choose an operator fˆ = f (n̂) which is a function
of the number operator n̂. For this choice all matrix elements hm|f (n̂)|ni with m 6= n
vanish, that is,
2
Fn ≡ ε0 hn|fˆ|ni
and
Cn ≡ 0 .
(8.14)
Hence, eq. (8.13) determines the exact solution
p
fˆ ≡ f (n̂) ≡ ~ ν0 ω ln(n̂ + 1)
which coincides with the definition of the Riemann Hamiltonian, eq. (3.2).
The answer to the question how one could realize this Hamiltonian experimentally
would exceed the scope of this thesis. Thus, we only mention here that Ruynet L. de
Matos Filho and Werner Vogel propose in [52] the ‘engineering of the Hamiltonian of a
trapped atom’, which generates Hamiltonians reminiscent of eq. (8.8).
In the remainder of this section, we show that we get a good approximation of the
Riemann zeta function without the exact Riemann Hamiltonian.
8.2.2. Approximate solution
Inspired by the non-linear Jaynes-Cummings-Paul model [53], where fˆ ∼ â + ↠, we now
choose fˆ as a function of the position operator
κ x̂ ≡ √
â + â†
2
or, more precisely, as a polynomial
fˆ ≡ f (x̂) =
µX
max
fµ x̂µ
(8.15)
µ=0
with the inverse length κ introduced in eq. (D.15).
73
8. Jaynes-Cummings-Paul approach to the Riemann Hamiltonian
We immediately see that in eq. (8.11) the absolute square of the matrix elements
|hn ± j|x̂µ |ni|2 only contain the elements with j ≤ µ, producing a polynomial in n with
degree µ (see appendix H.1 and H.2 for a general expression of the matrix elements).
This yields
(µmax )
Fn
µmax
2 2 ˆ 2 X (+)
ˆ
ˆ
εj
Θ(n − j) hn − j|f |ni + hn|f |n + ji
≡ ε0 hn|f |ni +
(8.16)
j=1
and likewise for eq. (8.12)
(µmax )
Cn
≡
µX
max
(−)
εj
j=1
2 2 ˆ
ˆ
Θ(n − j) hn − j|f |ni − hn|f |n + ji .
(8.17)
Both coefficients are polynomials in n of the order µmax . Here, the Heaviside function
Θ is defined by
(
1,x≥0
Θ(x) =
.
0,x<0
Since on the one hand there is no polynomial series expansion of the logarithm and, on
the other, a logarithmic dependence is always smaller than a linear one, one is inveigled
to assume that the best approximation of eq. (8.13) is given by the lowest terms of a
polynomial. However, our goal is to approximate the Riemann zeta function, that is to
compare the exact overlap C(t) ≡ hΦ|Ψ(t)i, eq. (3.6), with the approximation
hΦ|e
− ~i Ĥeff t
|Ψ(0)i =
∞ X
n=0
φ ne ce ψn e
− it
Fn + Cn
~
+ φ ng cg ψn e
it
~
Fn − Cn
.
(8.18)
Here, the effective Hamiltonian Ĥeff , eq. (8.10), is used to calculate the time evolution
of the initial state |Ψ(0)i. This task benefits of three properties of the overlap:
(i) The arguments of the exponential functions of both equations must only be equal
by modulo 2π, that is
t
!
Fn ± Cn mod 2π = ωt ln(n + 1) mod 2π .
~
(ii) The factors φ ne ψn and φ ng ψn are in the case of the zeta function proportional to
n−σ , which makes deviations of Ĥeff from ĤR for larger n negligible, and
(iii) deviations for one special n can be compensated by the others since only the sum
over all n matters.
Hence, even if the effective Hamiltonian does not provide a good approximation to the
Riemann Hamiltonian, the overlap can nevertheless approximate the zeta function quite
well. We show this behavior now exemplarily for the Riemann state |σ, τ i and a roughly
approximated Hamiltonian.
74
8.2. Riemann Hamiltonian
Riemann state
First, we recall that the Riemann state |σ, τ = ωti, eq. (4.6), is only defined for the
region σ > 1, where the zeta function is given by the Dirichlet series (4.2). Moreover,
the overlap reads
C(t) = hσ, 0|σ, τ i = |N (σ)|2 ζ(σ + iωt)
according to eq. (4.7).
Then, we derive from eq. (8.18) the overlap
C(t) = |N (σ)|2
∞
X
n=0
it
1
e− ~
σ
(n + 1)
Fn + Cn
.
Now, we have to choose the coefficients fµ in eq. (8.15) in such a way that
F≡
∞
X
n=0
it
1
e− ~
(n + 1)σ
Fn + Cn
!
=
∞
X
n=0
1
e −iωt ln(n+1) = ζ(σ + iωt)
(n + 1)σ
(8.19)
is approximately fulfilled. To simplify this task slightly more, we can reduce the number
of coefficients by choosing small atom frequencies. Indeed, for ωa Ω the largest
contribution
ˆ 2
ε
hn|
f
|ni
(8.20)
Fn(µmax ) ∼
= 0
is given by the inverse energy ε0 . A closer examination of the matrix elements (see
appendix H.1) shows, that only the even coefficients occur in hn|fˆ|ni. Hence, we choose
for the odd coefficients f2µ+1 = 0. This choice is also justified by the fact that all
contributions with odd coefficients are polynomials of the order 2µmax − 1, whereas the
even contributions are of order 2µmax in n.
Although eq. (8.20) is similar to the exact solution eq. (8.14), the coefficients Cn(µmax )
do not vanish here. On the contrary, Cn(µmax ) is also given by the even contributions with
order 2µmax in n, but it is smaller than Fn(µmax ) due to the missing inverse energy ε0 .
In fig. 8.2, we show eq. (8.19) for µmax = 4 and the coefficients f1 = f3 = 0, f0 = 0.2,
f2 = 0.3 and f4 = −0.4 for the Riemann state with σ = 2. The explicit expressions of
Fn(4) and Cn(4) are given in appendix H.3. In the upper row of fig. 8.2, we have depicted
the real and imaginary part of F in blue, eq. (8.19), and ζ(σ + iωt) in red. The overall
behavior fits quite well even though F fluctuates strongly. This becomes more evident
when we compare |F| to the approximation of the Dirichlet series
ζ(σ + iωt) ≈ 1 +
cos(ωt ln 2)
sin(ωt ln 2)
−i
≡ ζ̄ ,
2σ
2σ
(8.21)
indicated by the yellow curve in the picture below, to emphasize the periodic structure
of F.
In the third row, we illustrate the deviation
∆≡
|F|
|ζ(σ + iωt)|
(8.22)
75
8. Jaynes-Cummings-Paul approach to the Riemann Hamiltonian
Re F
Im F
1.6
0.4
1.4
0.2
1.2
20
1.0
40
60
80
100
t
-0.2
0.8
0.6
20
40
60
80
t
100
-0.4
|F|
1.6
1.4
1.2
1.0
0.8
0.6
∆
1.6
1.4
1.2
1.0
0.8
0.6
∆(n+)
3.0
2.5
2.0
1.5
1.0
0.5
20
40
60
80
100
t
Figure 8.2.: The real and imaginary part of F, eq. (8.19), is shown in the first row for Fn(4) in
blue. The parameters are chosen as follows: f0 = 0.2, f1 = f3 = 0, f2 = 0.3, f4 = −0.4, σ = 2,
ω = ωa = 1, Ω = 100, ~ = 1, κ = 1 (sum cut at N = 100). Although F fluctuates around the
values of ζ(σ + iωt) (red curve) the overall behavior fits quite well. This becomes evident in the
second row, where the periodic structure of |F| (blue) is given by the approximation of |ζ(2+ it)|,
eq. (8.21), indicated in yellow. Furthermore, the deviation ∆, eq. (8.22), (third row) between F
and the zeta function itself is, as expected, smaller than the deviation of the exponents ∆(n+) ,
eq. (8.23), shown in the bottom row.
76
8.3. Summary
of the approximation to the exact values. It is smaller than the deviation
∆n(+) ≡
(Fn + Cn ) mod 2π
~ω ln(n + 1)
(8.23)
of the exponents in eq. (8.19) depicted below. Hence, F provides a better approximation of the zeta function than the exponent Fn + Cn of the logarithm. Of course, the
approximation can be improved by using higher order polynomials and optimizing the
choice of the coefficients as well as the detuning between the atom frequency ωa and the
field frequency Ω.
8.3. Summary
In this chapter, we have shown that it is possible to get an approximation of the Riemann zeta function by using a Jaynes-Cummings-Paul-like model. Even if the effective
Hamiltonian resulting from this model does not match the Riemann Hamiltonian the
resulting overlap fits the overall behavior of the zeta function quite well, as the rough
approximation with a polynomial of fourth order for fˆ confirms.
A further investigation of approximations with higher order polynomials or other
functional dependence as well as the application of this method to other zeta states is
left for future work.
77
Summary
Due to its intimate connection to the distribution of the primes, the Riemann zeta
function ζ is famous beyond the borders of mathematics. Therefore, we have discussed
different representation of ζ in this thesis, adopting mathematical as well as physical
methods:
For the illustration of ζ in complex space, we have used the continuous Newton
method, which depicts a complex function F by lines of constant phase. These lines
start at the poles of the function and lead into its zeros. Hence, we can easily locate the
zeros and poles in the pictures of the Newton flow. Moreover, the zeros of the derivative
F 0 are visible since they generate a crossing of the flow lines which act as separatrices
for the flow into the different zeros. In case of the Riemann zeta function, the pictures
of the Newton flow would therefore reveal if the Riemann hypothesis was violated.
Our physical approach is inspired by the interaction of a cavity field with a twolevel atom, described by the Jaynes-Cummings-Paul model. We realize the Riemann
zeta function by the overlap hΦ|Ψ(t)i of two quantum states, but with a time evolution
governed by the Riemann Hamiltonian ĤR which depends logarithmically on the number
operator n̂. The properties of the states depend on the formula we want to reproduce:
In the region where the real part σ of the argument s is larger than one, the cavity
system suffices to produce ζ. However, entanglement is crucial for most representations
in the critical strip.
We have shown in particular that:
(i) The Dirichlet sum, defining ζ for σ > 1, is proportional to the overlap of the
Riemann state |σ, τ i with its initial state |σ, 0i. But the limitation of the Dirichlet
sum to the right of the critical strip is transferred to the states, due to conservation
of probability.
(ii) In contrast to the Dirichlet series, the alternating sum converges already for σ > 0.
However, our physical approach fails to enter the critical strip as long as we try
to reproduce the exact formula of the infinite alternating sum. The normalization
again restricts the states to σ > 1. Even entanglement is useless in this case.
(iii) Only the truncated version of the alternating sum allows the realization of ζ for
σ < 1 by a single system. The resulting truncated Riemann states |σ, τ iν are
normalizable as long as the truncation ν is finite. But when we increase the
truncation to improve the approximation of ζ by the truncated alternating sum,
the normalization of the states tends to zero and decreases the value of the overlap
as well.
79
Summary
(iv) Nevertheless, entanglement is the key to overcome the wall at σ = 1:
Unlike the Dirichlet series and the alternating sum, the approximations of ζ by
the Riemann-Siegel and the Berry-Keating formula involve not only the phases
−iτ ln(n + 1), but also the conjugate phases iτ ln(n + 1), which are available
through the entanglement with the atom. Moreover, the sums in these representations are truncated at relatively small limits n0 ≤ nB . Consequently, we need
only a few Fock states to produce these representations by the overlaps hΦRS|ΨRS(t)i
and hΦBK|ΨRS(t)i, respectively. The similarities and differences of the formulae are
reflected in the following properties of the states:
• The time-evolved entangled Riemann-Siegel state |ΨRS(t)i is used for both representations. It consists of two truncated Riemann states coupled to the excited and ground state of the atom. Thus, it is reminiscent of the Schrödinger
cat state, but with phases logarithmic instead of linear in n. Due to the entanglement, the time evolution acts in different ways on the matrix elements
of the Wigner representation Ŵ|ΨRSi : The diagonal matrix elements rotate
in opposite directions depending on the state of the atom. In contrast, the
off-diagonal matrix elements only change their internal structure, but remain
symmetric in the phase space variable p.
• The Riemann-Siegel reference state |ΦRSi involves the initial truncated Riemann states |σ, 0in0 and |2 − 3σ, 0in0 coupled to the atom. It is entangled for
σ 6= 1/2 and disentangled for σ = 1/2. Hence, the phase space pictures of the
state reveal a symmetry in p on the critical line.
• In contrast, the Berry-Keating reference state |ΦBKi contains the Berry states
|φB(σ, τ )inB and |φB(2 − 3σ, τ )inB and is entangled for all σ even though the
probability amplitudes differ only slightly from the ones of the initial truncated Riemann states used in |ΦRSi. Anyway, the off-diagonal matrix elements
of Ŵ|ΦRSi are symmetric in p for σ = 1/2.
Moreover, the special role of the critical line reveals itself in the phase-space representations of both approximations: The kernel K of the Wigner representation as
well as the absolute value of the corresponding kernel of the Moyal representation
K is p-symmetric for σ = 1/2.
(v) Additionally, we have shown that the Riemann zeta function can be simulated
by the time evolution with a non-logarithmic effective Hamiltonian: Already a
polynomial of fourth order produces a rough approximation of ζ reflecting the
overall behavior quite well although the Hamiltonians do not match.
We conclude this thesis by noting that our physical approach can be generalized to
other complex functions if they can be represented by the series
∞ X
an (σ, τ )
bn (σ, τ )
f (σ + iτ ) ≡
+
.
(n + 1) iτ
(n + 1)−iτ
n=0
80
As in the case of the Riemann zeta function, the factors (n + 1)±iτ can be interpreted as
time evolution of a quantum state |Ψ(t)i governed by the interaction Hamiltonian ĤR
and the rescaled time τ = ωt. The factors an and bn are distributed to the probability
amplitudes of the state |Ψ(t)i and the appropriately chosen reference state |Φi. But
it is crucial to keep in mind that the amplitudes of the initial state |Ψ(0)i must be
independent of τ . Otherwise we cannot reproduce the phases (n + 1)±iωt in |Ψ(t)i by
this simple time evolution. As a consequence, all τ -dependencies of an and bn must be
included in the probability amplitudes of the reference state, since there τ = Im s is
simply a constant determining the imaginary part of the point s in complex space.
In this way, we can transfer the characteristics of the complex function into properties
of quantum states and make them available by measurement.
81
A. Special functions
In this appendix, we bring back to mind some properties of the functions Γ, χ and ϑ,
which appear in the different representations of the Riemann zeta function.
A.1. The gamma function Γ
Already in 1729, different representations of the gamma function were given by Daniel
Bernoulli and Leonhard Euler in letters to Christian Goldbach [54]. We recall here only
the more familiar representations and properties of Γ.
A.1.1. Definition and functional equation
The gamma function Γ is defined by [50, 55]
Γ(s) ≡
Z∞
dx xs−1 e −x
(A.1)
0
for Re s > 0. Integration of this equation by parts yields the functional equation
Γ(s) =
1
Γ(s + 1)
s
(A.2)
which is used to define the analytical continuation of Γ in the rest of the complex plane.
From eq. (A.1) easily follows that Γ (s) = Γ(s̄) as well as the special values Γ(1) = 1
and Γ(1/2) = π 1/2 . With the help of eq. (A.2), we find the asymptotic expression
Γ(s) ∼
= s−1 for s → 0.
Another expression for the gamma function is the Gauß representation
Γ(s) = lim
n→∞
n! ns
.
s · (s + 1) · ... · (s + n)
It clearly shows that Γ has poles at zero and all negative integer numbers. This property
determines the location of the trivial zeros of the Riemann zeta function.
A.1.2. Asymptotics
For large τ we can approximate Γ with the method of stationary phase by
Γ(s) =
Z∞
0
dx e
−S(x)
∼
= e −S(xs )
Z∞
1
e− 2 S
00 (x
s) x
2
,
−∞
83
A. Special functions
where we have used the Taylor expansion of the phase S(x) = x − (s − 1) ln x around the
stationary point xs = s − 1 given by S 0 (xs ) = 0. Integration yields the Stirling formula
p
Γ(s) ∼
= 2π(s − 1) e −(s−1)[1−ln(s−1)]
for Re s > 1 which we can expand to smaller real parts with the help of the functional
equation (A.2). This results in the expression
√
1
Γ(s) ∼
(A.3)
= 2π e −s+(s− 2 ) ln s
for Re s > 0.
Moreover, we find from the series expansions [50]
x2
ln(1 + x) ∼
+ O(x3 )
−1 < x ≤ 1
=x−
2
∼ π − 1 + O x−3
x>1
arctan x =
2 x
for s = σ + iτ and large τ that
π σ ln s ∼
−
+ O τ −2 .
= ln τ + i
2
τ
(A.4)
Hence, eq. (A.3) can be expressed by
√
πτ
π
1
1
−1
∼
ln τ −
+i
σ−
− τ + τ ln τ + O τ
.
Γ(s) = 2π exp
σ−
2
2
2
2
(A.5)
A.1.3. Formulae
The gamma function is connected to the trigonometric functions [56] via
π
Γ(s) Γ(1 − s) =
sin(πs)
and
(A.6)
π
.
cos(πs)
(A.7)
22s−1
Γ(2s) = √ Γ(s) Γ (s + 12 ) .
π
(A.8)
Γ (s + 12 ) Γ (s − 12 ) =
For twice the argument we get
All three equations are used in the next section to show the equivalence of the different
representations of χ found in literature.
A.2. The function χ(s)
The function χ plays a crucial role in the analytic continuation of the Riemann zeta
function. It connects the values of ζ for σ > 1/2 to the region where σ < 1/2 via the
functional equation (1.5).
84
A.2. The function χ(s)
A.2.1. Definitions
In literature, χ is defined in many ways: While Siegel [29] used the formula
χ(s) ≡
(2π)s
2Γ(s) cos
πs
2
we find in [28] the representation
χ(s) ≡ π
and additionally in [4]
s− 21
Γ 1−s
2
Γ 2s
,
(A.9)
χ(s) ≡ 2s π s−1 Γ(1 − s) sin
(A.10)
πs 2
.
(A.11)
The equivalence of representation (A.11) and (A.9) becomes evident when we use
eq. (A.6) to replace the sine. To arrive at definition (A.10), we substitute Γ(s) in
eq. (A.9) using the doubling formula (A.8) and apply eq. (A.7).
The function χ inherits the property χ (s) = χ(s̄) of the Gamma function, as eq. (A.10)
easily confirms. However, the functional equation reads
χ(1 − s) = χ−1 (s) .
(A.12)
Moreover, eq. (A.10) shows that there exist only ‘trivial’ zeros of χ, which are located
at s = 0, −2, −4, ... due to the poles of the denominator Γ(s/2), and poles at s = 1, 3, 5, ...
created by the poles of the numerator Γ((1 − s)/2).
A.2.2. On the critical line
Already the functional equation (A.12) of χ indicates the special role of the critical line.
Indeed, there we find
1 iτ
χ ( 12 + iτ ) = exp i τ ln π − 2 arg Γ
+
,
4
2
that is |χ ( 21 + iτ ) | = 1 for all τ . With the help of eq. (A.3), we get the approximation
n τ
π o
χ ( 21 + iτ ) ∼
−τ −
+ O(τ −1 )
= exp −i τ ln
2π
4
(A.13)
for large τ .
85
A. Special functions
A.2.3. Asymptotics in the critical strip
In the critical strip 0 < Re s < 1, we find that both gamma functions in the definition
(A.10) of χ have arguments with real part between 0 and 1/2. Hence, we can approximate
them for τ 1 with the help of eq. (A.3) and the expressions
s
2
1−s
ln
2
ln
τ
iπ
iσ
∼
−
+ O(τ −2 )
= ln +
2
2
τ
τ
iπ
i(1 − σ)
∼
−
+ O(τ −2 )
= ln −
2
2
τ
for the logarithms. Eventually, we arrive at [4, 5]
σ− 1
2
2π
t
π
−1
χ(s) ∼
+ O(τ )
exp −i τ ln
−τ −
=
τ
2π
4
(A.14)
which is according to eq. (A.13)
χ(s) ∼
=
2π
τ
σ− 1
2
χ( 21 + iτ ) .
(A.15)
A.3. The function ϑ(s)
In the derivations of the Riemann-Siegel and Berry-Keating formula in appendix B, we
use the definition
e i ϑ(s) ≡ χ−1/2 (s)
(A.16)
to express the symmetric representation Z, eq. (1.7). This yields
"
1−s #
h s i
Γ
1
1−s
i
1
ϑ(s) ≡
arg Γ
− arg Γ
+
s−
ln π + ln 2s 2
2
2
2
2
Γ 2
and for σ = 1/2
τ
1 iτ
θ(τ ) ≡ ϑ( 21 + iτ ) = arg Γ
+
− ln π
4
2
2
(A.17)
which is real. Since Z is also real for s = 1/2 + iτ , θ(τ ) describes the phase of ζ on the
critical line. This phase is, according to [9], closely related to the phase shift produced
by scattering on an inverted harmonic oscillator.
Besides the critical line, the complex conjugate of ϑ fulfills
ϑ (s) = −ϑ(s̄)
since the property
Γ (s) e −i arg [ Γ(s) ] = Γ (s) = Γ(s̄) = |Γ(s̄)| e i arg [ Γ(s̄) ]
results in
86
Γ (s) = |Γ(s̄)|
and
arg [ Γ(s) ] = − arg [ Γ(s̄) ] .
B. Approximations in the critical strip
The exact formulae of the zeta function are either not capable of producing states that
could enter the critical strip, chapter 4 and 5, or are difficult to realize experimentally.
Thus, we make use of the Riemann-Siegel and the Berry-Keating formula which are quite
good approximations for large imaginary parts τ and can be conveniently casted in our
physical picture.
In this appendix, we give at first a short review of the Riemann-Siegel formula with
focus on the remainder to provide the necessary expressions for the following sections.
Then, we show that the Berry-Keating formula on the critical line is an improved version
of the Riemann-Siegel formula since it contains additionally its first correction term.
Finally, we give the extension of the Berry-Keating formula to the critical strip.
B.1. Riemann-Siegel formula
As mentioned in section 1.5, Carl L. Siegel rederived the semi-convergent representation
of the zeta function [29]
ζ(s) = ζRS(s) + RS
(B.1)
which contains in the main part
ζRS(s) ≡
n0
n0
X
X
1
1
+
χ(s)
s
1−s
n
n
n=1
(B.2)
n=1
the function χ(s) defined by eq. (1.6). Due to its dependence on τ , the cut-off n0 ,
eq. (1.10), leads to discontinuities.
For the sake of completeness, we note that the remainder is according to [29]
s+1
(−1)n0 −1 (2π) 2 τ
RS ≡
(1 − e 2πis )
Γ(s)
with
SN ≡
N
−1
X
s−1
2
e i[
πs
− τ2 − π8
2
]S
N
N !
X 2−k i r−k k!
6
3N
ak F (k−2r) (δ) + O
.
r! (k − 2r)!
τ
k=0 r≤k/2
The summation limit is N ≤ 2 · 10−8 τ and the derivatives F (k) of
cos u2 + 3π
8
√
F (u) ≡
cos 2πu
87
B. Approximations in the critical strip
are evaluated at δ ≡
recurrence formula
√
τ−
√
2π n0 +
1
2
. Moreover, the coefficients ak in SN obey the
√
(n + 1) τ an+1 = −(n + 1 − σ) an + i an−2
with a−2 = a−1 = 0 and a0 = 1. In literature [4, 29, 31], we find various representations
of the whole remainder, which differ slightly due to their derivation. We will concern
ourselves in what follows only with the first correction term
s+1
(0)
RS (s)
with
(−1)n0 −1 (2π) 2 τ
≡
(1 − e 2πis )
Γ(s)
s−1
2
e i[
πs
− τ2 − π8
2
√
cos τ − (2n0 + 1) 2πτ − π8
√
F ≡
cos( 2πτ )
]F
(B.3)
given by Siegel [29] for τ > 0. The remainder for negative imaginary part τ follows from
the symmetric form Z(s), eq. (1.7).
When we take the Riemann-Siegel representation (B.1) with the remainder approxi(0)
mated by its first correction term RS to evaluate Z, we arrive at
n0
X
2
1
(0)
(0)
√ cos ϑ(s) + i s −
ZRS (s) ≡
ln n + χ−1/2 (s) RS (s) ,
(B.4)
2
n
n=1
where ϑ is defined by eq. (A.16). Since Z is symmetric to the critical line, eq. (1.8), the
remainder fulfills
(0)
(0)
RS (1 − s) = χ−1 (s) RS (s)
in analogy to the functional equation of the Riemann zeta function, eq. (1.5). Hence,
(0)
R(0) (s) ≡ χ−1/2 (s) RS (s)
has to be real on the critical line. Moreover,
1/4
2π
(0)
R (s) ∼
(−1)n0 −1 F
=
τ
(B.5)
holds in the whole critical strip, as the approximations of Γ and χ, eqs. (A.5) and (A.14),
for τ 1 reveal. In the next section, we show that the representation of Z given by
Berry and Keating combines the main term and the first correction R(0) in the first
summand of a series expansion.
B.2. Berry-Keating formula on the critical line
Michael V. Berry and Jonathan P. Keating show [31] that one can cast the symmetric
representation on the critical line
Z ( 12 + iτ ) = e iθ(τ ) ζ( 12 + iτ ) ,
88
B.2. Berry-Keating formula on the critical line
with θ given by eq. (A.17), into a series of convergent sums
Z ( 12 + iτ ) = Z0 (τ, K) + Z3 (τ, K) + Z4 (τ, K) + . . . .
Here, K is an arbitrary constant and the main part Z0 can be expressed by
(∞
)
r
X exp {i(θ(τ ) − τ ln n)} 1
τ
√
Erfc ξ(n, τ )
Z0 (τ, K) ≡ 2 Re
2
2Q2 (K, τ )
n
n=1
(B.6)
with the abbreviations
ξ(n, τ ) ≡ ln n − θ0 (τ )
and Q2 (K, τ ) ≡ K 2 − iτ θ00 (τ )
(B.7)
as well as the complementary error function (eq. (7.1.2) in Abramowitz & Stegun [56])
2
Erfc (x) ≡ √
π
Z∞
2
dt e −t .
(B.8)
x
Following the paper, we first show that the sum over n is convergent and then prove
(0)
that the main term Z0 is approximately ZRS , eq. (B.4).
In eq. (B.6), the convergence of the sum is based on the property that Erfc (x) is for
2
large arguments proportional to e −x . Indeed, with
r
τ
(B.9)
y(n, τ, K) ≡ ξ(n, τ )
2
2Q (K, τ )
we find


!2 
!2 




2
n
n
τK
τ
p
p
ln
ln
=
exp
−
Erfc
(y)
∼
exp
−
τ
τ


 2(K 4 + 14 )
 2K 2 − i
2π
2π
which becomes exponentially small for large τ . Hence, we can choose > 0 to fulfill the
inequality

!2 


2
τK
n
p
exp −
ln
≤ e − .
τ
 2(K 4 + 14 )

2π
This yields
n≤
r
τ
exp
2π
(
s )
1
1+
π
4K 4
2π
K
pτ
(B.10)
√
for the summation
p index. For K = 1/ 2, the expression on the right is minimal and
the exponent 2ε/τ vanishes for large τ . Therefore, it suffices to cut the sum at n0 ,
eq. (1.10), to get the main contribution.
Now, we use the approximation (A.13) of χ to get the approximations for
h
i τ
τ
τ
π
θ(τ ) = arg χ−1/2 (1/2 + iτ ) ∼
− − + O(τ −1 )
(B.11)
= ln
2 2π
2
8
89
B. Approximations in the critical strip
and for its derivatives
1
τ
θ0 (τ ) ∼
+ O(τ −2 )
= ln
2 2π
and
1
θ00 (τ ) ∼
+ O(τ −3 ) ,
=
2τ
which transform eq. (B.7) into
n
ξ∼
= ln p τ
and
2π
i
Q2 ∼
= K2 − .
2
(B.12)
Hence, eq. (B.9) is approximately
!r
i
i
n
τ
arctan 1 2
arctan 1 2
2
2
2K
2K
y∼
e
≡
y
e
.
= ln p τ
0
2
2|Q|
2π
Taking into account that the complementary error function with complex argument
is roughly given by1
(B.13)
Erfc (y0 e iφ ) ≈ Erfc (y0 )
as long as |φ| ≡ 21 arctan 2K1 2 ≤ π4 , which holds for all K, we arrive at the approximation
(∞
)
X exp {i(θ(τ ) − τ ln n)} 1
√
Z0 (τ, K) ∼
Erfc(y0 )
= 2 Re
2
n
n=1
for the main part, eq. (B.6). Furthermore, the relation
1
1
Erfc (y0 ) = Θ(−y0 ) + Erfc (|y0 |) sgn(y0 ) ,
2
2
where the Heaviside function and the signum are defined by




0
,
x
<
0


 −1 , x < 0

1
Θ(x) ≡
and
sgn(x) ≡
0, x=0
2 , x=0




 1, x>0
 1, x>0
produces
Z0 ∼
=2
1
,
(B.14)
n0
X
cos[θ(τ ) − τ ln n]
√
+R.
n
n=1
When we calculate the integral in definition (B.8) along the path t = r e iφ , we arrive at
2
Erfc(y0 e iφ ) = √ e iφ
π
Z∞
dr exp{−r2 cos(2φ) − ir2 sin(2φ)}
y0
which converges only if |φ| ≤ π/4.
eq. (B.13).
√ The rough approximation cos x 1≈ 1 produces
arctan 1 2 = π/8 and
At
the
end,
we
choose
K
=
1/
2
for
which
the
phase
is
|φ|
=
2
2K
|Erfc (y0 e iπ/8 )| − Erfc (y0 ) ≤ 0.07.
90
B.2. Berry-Keating formula on the critical line
Here, we have already used that the Heaviside function Θ(−y0 ) cuts the sum at n0 to
reveal the main part in eq. (B.4). The remainder
(∞
)
X exp {i(θ(τ ) − τ ln n)}
√
R ≡ Re
Erfc(|y0 |) sgn(y0 )
(B.15)
n
n=1
becomes quite small at the jumps τ = 2πn20 of the Riemann-Siegel formula.
When we substitute n = n0 + k and τ = 2π(n0 + p)2 in eq. (B.12) and approximate
the logarithm in ξ according to eq. (A.4), we get
n
k−p ∼ k−p
1 k−p 2
p
ξ∼
ln
=
ln
1
+
−
=
=
τ
n0 + p
n0 + p 2 n0 + p
2π
and can estimate the argument of the exponential function in eq. (B.15) by
"
#
h
i
n
τ
π
π
θ(τ ) − τ ln n mod 2π ∼
mod 2π ∼
= −τ ln p τ − −
= πn0 + 2πp2 − + πk − 4πpk ,
2
8
8
2π
using additionally the approximation of θ, eq.√ (B.11). From the above substitution of τ
π
and the approximation of ξ follows that y0 ∼
= |Q| (k −p). Therefore, eq. (B.15) transforms
into
(
1
4
1
2π
2
R∼
(−1)n0 Re e 2πi(p − 16 )
=
τ

√

√
∞
X
π|k − p|
π(k − p)
(B.16)
sgn
·
(−1)k e −i4πpk Erfc

|Q|
|Q|
k=1−n0
−1/2 ∼
when we additionally apply n−1/2 ∼
= (2π/τ )1/4 . Since the error function vanishes
= n0
for large k, we can extend the sum to −∞ and split it into positive and negative k. For
p > 0, that is between the discontinuities at τ = 2πn20 , eq. (B.16) yields
∞
X
|k − p| ∼
k
−1 − 2i
(−1) sin(4πpk) Erfc
= −1 + i tan(2πp) .
|Q|
k=1
Here, we have used that the signum, eq. (B.14), becomes +1 for k > 0 and −1 for k ≤ 0.
In the last step, we have assumed that the error function can be estimated by 1. This
rough approximation is justified by the fact that 0 < p < 1 and the overall factor τ −1/4
makes R small for large τ . Finally, after some calculation and with the help of eqs. (B.3)
and (B.5), we get that the remainder
1
1/4
1
2
2π 4
2π
n0 −1 cos[2π(p − p − 16 )]
∼
R=
(−1)n0 −1 F ∼
(−1)
=
= R(0) (1/2 + iτ )
τ
cos(2πp)
τ
is approximately the first correction term R(0) of the Riemann-Siegel formula.
Additional to the improvement of the approximation, the Berry-Keating representation benefits from several more properties [31]:
91
B. Approximations in the critical strip
(i) The discontinuities of the Riemann-Siegel formula are eliminated since all terms
are analytic functions of τ .
(ii) The other terms Zi , i ≥ 3, ‘can be estimated explicitly’ and
(iii) ‘numerical studies suggest that term by term the new series is more accurate than’
the Riemann-Siegel expression.
Therefore, we present the extension of the Berry-Keating formula to σ 6= 1/2 in the
following section.
B.3. Berry-Keating formula in the critical strip
Since we are also interested in a description of the zeta function besides the critical line,
we now extend the Berry-Keating formula to the critical strip. Therefore, we first bring
eq. (B.6) into the form
Z0 =
∞
X
e i(θ(τ )−τ ln n) Erfc(y) + e −i(θ(τ )−τ ln n) Erfc (y)
√
2
n
n=1
with y ≡ y(n, τ, K) given by eq. (B.9). Then, we introduce the argument s ≡ σ + iτ
by
1
substituting the function θ(τ ) with ϑ(s), eq. (A.16), and τ itself with T ≡ i 2 − s :
Z0 (s) =
∞
X
e i(ϑ(s)−T
n=1
ln n)
Erfc(y(n, T , K)) + e −i(ϑ(s)−T
√
2 n
ln n)
Erfc (y(n, T , K))
.
We recall the approximation (A.15) of χ in the critical strip and arrive with the
definition (A.16) at
1 σ
2π
ϑ(s) ∼
θ(τ
)
−
i
−
ln
, ϑ0 (s) ∼
=
= θ0 (τ ) and ϑ00 (s) ∼
= θ00 (τ ) ,
4 2
τ
where we have neglected all terms smaller or equal to O(τ −1 ). These approximations
yield
ξ(n, T ) ≡ ln n − ϑ0 (s) ∼
= ξ(n, τ )
as well as
Q2 (K, T ) ≡ K 2 − iT ϑ00 (s) ∼
= K 2 − it θ00 (τ ) + O(τ −1 ) ∼
= Q2 (K, τ )
for eq. (B.7), and thus
y(n, T , K) ∼
= y(n, τ, K)
for the argument of the error function. Hence, the same steps as in the previous section
(0) (s), eq. (B.4).
can be applies to proof that ZBK(s) is approximately ZRS
92
B.3. Berry-Keating formula in the critical strip
√
Finally, we take advantage of eq. (B.10) by choosing K = 1/ 2 as well as the truncation parameter nB ≥ n0 and use the approximations in eq. (B.12) to get
"
#
nB
X
χ−1/2 (s) E(n − 1, τ ) χ1/2 (s) E (n − 1, τ )
ZBK(s) ≡
+
,
(B.17)
ns
n1−s
n=1
where
1
E(n, τ ) ≡ Erfc
2
"r
τ
n+1
ln p τ
1−i
2π
#
.
Since ZBK provides an excellent approximation to the symmetric representation Z (see
figures 1.2 and 1.3), we use eq. (B.17) to derive the Berry-Keating reference state in
chapter 7.
93
C. Continuous Newton method
Partial differential equations govern many physical phenomena. Their solution can often
be found by casting the problem into a search for the minimum of a function. Hence,
continuous descent methods [39, 40], like the continuous Newton method, have found
wide application in physics.
In this appendix, we illustrate the building blocks of the Newton flow by means of
several examples. The last sections are dedicated to technical details needed for the
pictures and to describe the Newton flow of ζ.
C.1. Newton flow
As already mentioned in section 2, the continuous Newton method consists of finding
the solution s(t) for the differential equation
ṡ(t) ≡
ds(t)
F (s(t))
=− 0
.
dt
F (s(t))
(C.1)
Provided that F 0 (s) ≡ dF (s) 6= 0, eq. (C.1) yields
ds
F 0 (s(t)) · ṡ(t) =
dF (s(t))
= −F (s(t))
dt
for a holomorphic function F . Hence, the trajectories s(t) are defined by
F (s(t)) = F (s(0)) e −t
(C.2)
and represent lines of constant phase. The totality of the Newton trajectories s(t) form
the Newton flow. We investigate the structure of the flow by some simple examples in
the next sections.
C.1.1. Sink and source
In the limit t → ∞, eq. (C.2) becomes zero. Therefore, all trajectories of the Newton flow
end eventually at a zero of F , which makes the continuous Newton method a perfect tool
to find the zeros of a function. Moreover, the limit t → −∞ indicates that the starting
point of the trajectories is a pole. We have to mention here that the direction of the
Newton flow results from the choice of the minus sign in the differential equation (C.1),
which is a mere convention. Choosing plus instead of minus would reverse the direction
of the flow, but not change the location of the curves in complex space.
95
C. Continuous Newton method
Figure C.1.: Newton flow of the functions F0 , Fp and the Möbius transformation FM defined in
eq. (C.3) (from left to right). The zero (red dot) acts as sink while the pole (black dot) is source
of the Newton flow. Infinity can be source and sink.
In fig. C.1 from left to right, we present the Newton flow of the functions
F0 (s) ≡ s ,
Fp (s) ≡ s−1
and
FM (s) ≡
s+1
s−1
(C.3)
in complex space s = σ + iτ . Here, F0 and Fp display a single zero and a single pole,
respectively, while the Möbius transformation FM possesses both. As expected, the zero
acts as a sink and the pole as a source for the Newton flow. Moreover, infinity can be
source and sink. Here and in all other pictures of the Newton flow, zeros and poles are
marked by red and black dots, respectively.
We note that although these pictures are reminiscent of the field of charges, there is
no one-to-one correspondence between the Newton flow and electrostatics [57]. Indeed,
if we consider the components N1 ≡ −Re {F/F 0 } and N2 ≡ −Im {F/F 0 } of the Newton
~ as components of an electric field, the component
vector N
~ ≡ ∂N2 − ∂N1
curl N
∂σ
∂τ
(C.4)
of the curl does in general not vanish due to the Cauchy-Riemann differential equations,
which are fulfilled by a holomorphic function. We give a short proof of this statement
before we turn to further examples in the next section.
Proof. Since F is holomorphic, the function −F/F 0 is also holomorphic and therefore
fulfills the Cauchy-Riemann differential equations [50]
∂N1 (σ, τ )
∂N2 (σ, τ )
=
∂σ
∂τ
and
∂N1 (σ, τ )
∂N2 (σ, τ )
=−
.
∂τ
∂σ
~ to simplify the notation.
Here, we have used the components of the Newton vector N
Hence, we get
~ ≡ 2 ∂N2
curl N
(C.5)
∂σ
from eq. (C.4), which in general does not vanish.
96
C.1. Newton flow
It is worthwhile to represent the curl of the Newton flow in terms of the real and
imaginary part u(σ, τ ) and v(σ, τ ) of the complex function F (s) = F (σ + iτ ): With the
help of the identity
dF (s)
∂F dσ
F 0 (s) =
=
= uσ + ivσ ,
ds
∂σ ds
where fσ denotes the partial derivative of the function f with respect to σ, the Newton
quotient reads
F
uuσ + vvσ
uvσ − uσ v
− 0 =− 2
−i 2
= N1 + iN2 .
F
uσ + vσ2
uσ + vσ2
Hence, the curl of the Newton flow, eq. (C.5), is given by
2u uσ vσ + v(vσ2 − u2σ ) uσσ + −2v uσ vσ + u(vσ2 − u2σ ) vσσ
~
curl N = −2
(u2σ + vσ2 )2
(C.6)
which illuminates the fact that the curl vanishes particularly at the zeros of F or of the
~ vanishes on the whole real axis if F (σ) = u(σ, 0) or on
derivative F 0 . Moreover, curl N
the whole imaginary axis if F (iτ ) = iv(0, τ ). We give now a short proof of the last two
statements.
Proof. The proofs are based on the property that the partial derivative of a function
f (x, y) with respect to one of the independent variables fulfills
∂f (x, y) ∂f (x, 0)
=
.
(C.7)
∂x y=0
∂x
(i) If F (σ) = u(σ, 0), then follows that the imaginary part v(σ, 0) = 0 and therefore
the derivatives vσ (σ, 0) and vσσ (σ, 0) are zero due to eq. (C.7). Hence, all terms in
the numerator in eq. (C.6) vanish on the real axis.
(ii) For F (iτ ) = iv(τ ), eq. (C.7) yields uτ (0, τ ) = 0 and uτ τ (0, τ ) = 0. Additionally,
we get uτ = −vσ and uτ τ = −uσσ from the Cauchy-Riemann differential equations.
~ vanishes on the imaginary axis.
Thus, curl N
C.1.2. Origin of separatrices
The structure of the Newton flow in the examples above can also explain the flow of more
complicated functions like higher order polynomials [41] or the Riemann zeta function.
Indeed, the flow lines of the polynomials
Fd (s) ≡ s2 − 1
and Ft (s) ≡ s3 − 1
(C.8)
shown in fig. C.2 all start at infinity and end in one of the zeros sd1/2 = ±1 or
stk = e 2πi k/3 , k = 0, 1, 2, of Fd or Ft , respectively. Yet, there have to be lines separating the flow towards the different zeros. In the case of the function Fd (left picture
97
C. Continuous Newton method
Figure C.2.: Newton flow of the functions Fd (left) and Ft (right), eq. (C.8). The separatrices,
indicated by the green dashed lines, separate the flow into the zeros sd1/2 = ±1 or stk = e 2π i k/3 ,
k = 0, 1, 2, of Fd or Ft , respectively. They cross each other in the origin, marked by a green
point, where the first derivative of the functions vanishes. Since Fd (0) = −1 = Ft (0), the phase
on the separatrices is π in these examples.
of fig. C.2), these lines are obviously given by the imaginary axis on which the flow is
always directed to the origin. There, it looses its forward uniqueness and can either
continue to the zero at +1 or to the zero at −1. Hence, we call such a crossing point
hyperbolic point [41] and refer to the crossing trajectories as separatrices.
We proof now:
(i) The Newton flow lines can only cross at points s0 where the first derivative F 0 (s0 )
vanishes, and
(ii) the number of separatrices through one of these hyperbolic points is determined
by the degree of the first non-vanishing derivative of the function.
Proof (i). Suppose, s0 is a hyperbolic point and F 00 (s0 ) 6= 0. Then,
1
2
F (s) ∼
= F (s0 ) + F 0 (s0 ) (s − s0 ) + F 00 (s0 ) (s − s0 )
2
is the Taylor expansion up to second order at this point. Using eq. (C.2) on the left
hand side yields the approximate behavior
s
0 (s )
F
F 0 (s0 ) 2
F (s0 )
0
∼
s1/2 (t) = s0 − 00
±
− 2 00
(1 − e −t )
00
F (s0 )
F (s0 )
F (s0 )
of two trajectories around s0 ≡ s(t0 ). Since s0 is the crossing point of the two trajectories,
s1 (t0 ) = s2 (t0 ) = s0 must be fulfilled which requires that the square root and the fraction
F 0 /F 00 vanish independently at t = t0 . Hence, the first derivative F 0 (s0 ) must be zero
and t0 = 0.
Proof (ii). Suppose, that m > 1 is the order of the first non-vanishing derivative of F .
Then, the Taylor expansion around the hyperbolic point s0 reduces to
1 (m)
F (s) ∼
F (s0 ) (s − s0 )m
= F (s0 ) +
m!
98
C.2. Further examples
and the approximate behavior of the separatrices is given by
2πi k/m
∼
m!
sk (t) = s0 + e
F (s0 )
F (m) (s0 )
1−e
−t
1
m
(C.9)
with k = 0, . . . , m − 1. Hence, there are m separatrices crossing in s0 .
Since the point s0 with F 0 (s0 ) = 0 in eq. (C.9) is reached at t = t0 = 0, negative t
reflect the incoming part of the separatrix whereas positive t mark the outgoing part.
Investigating the near neighborhood of the point s0 , that is |t| 1, we find that the
separatrices, eq. (C.9), are approximately given by
where
F (s0 ) 1/m 1/m
iφ
2πi k/m
∼
m! (m)
sk (t) = s0 + e · e
·t
,
F (s0 ) Im F (s0 )/F (m) (s0 )
1
.
φ≡
arctan
m
Re F (s0 )/F (m) (s0 )
Thus, the outgoing parts of the separatrices leave s0 under the angles φ + 2πk/m while
the incoming parts enter with φ + 2πk/m + π/m, because t = −|t| there.
In fig. C.2, the separatrices are straight lines, confirming the properties of separatrices
discussed before. Since the origin (green dot) is the only point where the derivatives
Fd0 and Ft0 vanish, the phase of the separatrices is π, indicated by dashed green lines.
Moreover, we see that the incoming parts of the separatrices divide the flows into different
zeros whereas the outgoing parts separate flows from different directions.
We emphasize that separatrices are not necessarily real. Recalling that Newton trajectories are lines of constant phase, it is obvious that the phase of the separatrix is
defined by the phase of the function F at the hyperbolic point on this line. Therefore,
we mark hyperbolic points and the corresponding separatrices in green.
It remains to mention that zeros with multiplicity larger than one act like simple zeros.
Although the first derivative vanishes at the zero, it does not create a separatrix. Indeed,
the solution of s0 (t) = s(0) e −t of eq. (C.1) for F0 ≡ s differs only by a factor 1/2 in the
exponential from the solution sd0 (t) = s(0) e −t/2 for Fd0 ≡ s2 . Hence, the Newton flow
approaches a zero with multiplicity 2 with half of the speed.
C.2. Further examples
This section is dedicated to the description of the Newton flow of the functions χ and
Fe which govern the asymptotic behavior of the Newton flow of ζ shown in section 2.2.
Since the Newton flow of both functions is symmetric to the real axis, we restrict the
description to the upper half of the complex plane.
99
C. Continuous Newton method
Figure C.3.: The left picture shows the Newton flow of the function χ, eq. (C.10). The separatrices through the hyperbolic points on the real axis have phase 0 (solid green lines) and π
(dashed green lines) in alternating sequence. The separatrix through the non-trivial zero of χ0 at
σ = 1/2 and τ ∼
= 6.23 consists of curves with phase ∼ π/4 (dot-dashed green lines) and divides
the complex space into the different regions listed in section C.2.1. The violet lines are additional
real lines with positive (solid) or negative (dashed) values and the orange line indicates σ = 1/2.
The right picture shows the Newton flow of the function Fe , eq. (C.11). There, all flow lines
start at −∞ and end eventually at a zero with s = ±i(2k + 1)π/ ln 2. The incoming parts of
the separatrices cross the complex plane parallel to the real axis, ending in the zeros of Fe0 at
+∞ with τ = ±i2k π/ ln 2, marked by green circles. They are positive real lines since Fe = 1 at
the hyperbolic points. The fact that the only lines which start at +∞ are also real and positive
suggest that they are the outgoing trajectories of the hyperbolic points. Therefore, we have
marked them in green with green arrows emphasizing the exceptional starting point at +∞.
We show only the upper part of the complex plane in the pictures, since the Newton flow is
in both examples symmetric with respect to the real axis.
100
C.2. Further examples
C.2.1. The function χ
As recalled in appendix A.2, the function
1
π s− 2 Γ 1−s
2
χ(s) ≡
Γ 2s
(C.10)
has zeros at the origin and the negative even integers whereas the poles are located
at the positive odd integers. Moreover, a hyperbolic point appears between every two
neighboring zeros or poles on the real axis and an additional hyperbolic point is located
on the critical line σ = 1/2 at τ ∼
= 6.23, which we call non-trivial zero of χ0 in analogy
to the non-trivial zeros of ζ.
The left picture of fig. C.3 shows the Newton flow of χ. There, the dot-dashed green
line marks the non-trivial separatrices with phase ∼ π/4 through the non-trivial zero of
χ0 . They divide the Newton flow in four regions:
(i) On the left of the non-trivial zero, the flow lines from −∞ end in the trivial zeros
with s0 ≤ −4, while
(ii) on the right, the ones ending at +∞ start at the poles with sp ≥ 5.
(iii) The flow below the outgoing non-trivial separatrices, that is around the origin,
connects the pole sp = 1 to the zero s0 = 0 as well as sp = 3 to s0 = −2 and sp = 5
to s0 = −4.
(iv) Above the non-trivial separatrix, the flow crosses the complex plane from −∞ to
+∞.
Additionally, the separatrices through the hyperbolic points on the real axis are indicated
by solid and dashed green lines for phase 0 or π, respectively. They direct the flow
in region (i) into the different zeros and separate the flow from the different poles in
region (ii). In region (iii), the separatrices connect the hyperbolic points σ ∼
= 1.45 and
σ∼
= −0.45 (phase π) as well as σ ∼
= 3.75 and σ ∼
= −2.75 (phase 0). Hence, they enclose
the flow between the pole and the corresponding zero sp = 1 and s0 = 0 or sp = 3 and
s0 = −2.
Finally, we note that there are no separatrices in region (iv). We have marked there
the real lines in violet (dashing indicates again the phase π) to emphasize their difference
to the separatrices of ζ discussed in section 2.2.2.
C.2.2. The function Fe
The function
Fe (s) ≡ 1 + e −s ln 2
(C.11)
contains the first two summands of the Dirichlet series of the Riemann zeta function,
eq. (4.1). It possesses zeros at ±i(2k + 1) π/ ln 2, k ∈ N, and zeros of Fe0 for σ → +∞
and τ = ±i2k π/ ln 2, indicated by green circles in the right picture of fig. C.3.
101
C. Continuous Newton method
The picture shows that all Newton flow lines start at −∞ and eventually end in a zero
of Fe . However, it seems that the green lines are exceptions since they are parallel to the
real axis. Yet, we can identify them as incoming and outgoing parts of the separatrices:
The incoming parts are obviously the green lines starting at −∞. They lead straight
into a zero of Fe0 at +∞ and separate the flows into neighboring zeros. Since Fe = 1 in
the hyperbolic points, the phase of the separatrices is zero.
The other green lines start at +∞, but already at the same height as the zero of Fe in
which they end. Nevertheless, the phase vanishes on these lines and they separate the
flow above the zero from the one below. Hence, we refer to them as the outgoing part
of the separatrices. Since they are the only lines which come from +∞, we have marked
them by green arrows.
C.3. Technical details
The pictures of the Newton flow are produced with Mathematica [47] and the commands
[41]:
F [s ] := ...
N 1[σ , τ ] := −Re[F [σ + Iτ ]/F 0 [σ + Iτ ]]
N 2[σ , τ ] := −Im[F [σ + Iτ ]/F 0 [σ + Iτ ]]
StreamPlot[{N 1[σ, τ ], N 2[σ, τ ]}, {σ, σ0 , σ1 }, {τ, τ0 , τ1 }]
for the intervals σ ∈ [σ0 , σ1 ] and τ ∈ [τ0 , τ1 ]. Here, I defines the imaginary unit i. The
zeros, poles and hyperbolic points are marked afterwards.
C.4. Newton flow of ζ
Figure C.4 shows the Newton flow of the Riemann zeta function up to an imaginary
part of τ = 300. The separatrices are depicted in green and additional real curves in
violet to emphasize their difference to the (real) ‘bones’ analyzed by Arias-de Reyna in
his ‘x-ray’ [36]. Solid and dashed lines indicate phase 0 and π, respectively, whereas the
phase of dot-dashed curves is determined by the hyperbolic point in which they cross.
The red dots mark the zeros of ζ [58, 59] and green dots the hyperbolic points s00 listed
in tables C.1 - C.3. All values have been calculated with Mathematica [47].
C.5. Zeros of the derivative ζ 0
In tab. C.1, we present the trivial zeros of the derivatives ζ 0 and χ0 . The difference
between the values decreases for zeros with smaller real parts.
Table C.2 contains the hyperbolic points s00 of the Riemann zeta function and the
phase at these points. Since the phase is quite small – the absolute values are smaller
102
C.5. Zeros of the derivative ζ 0
Figure C.4.: Newton flow of ζ for imaginary parts up to τ = 300.
103
C. Continuous Newton method
than 0.37 < 0.12 π for s00 with imaginary part up to τ = 100 – the non-real separatrices
of ζ are close to its real lines. This explains the ‘attraction’ of the real lines by the
hyperbolic points observed in [36].
Finally, tab. C.3 lists the hyperbolic points s00 with imaginary parts 100 < τ < 300,
used in fig. C.4.
s00
s0χ
-2.7173
-2.7482
-4.9368
-4.9425
-7.0746
-7.0758
-9.1705
-9.1707
Table C.1.: The trivial zeros s00 of the derivative ζ 0 of the Riemann zeta function are located
slightly to the right of the trivial zeros s0χ of χ0 . However, the difference between the values
vanishes for σ → −∞.
s00
arg{ζ(s00 )}
s00
arg{ζ(s00 )}
2.4632 + i 23.298
0.03317
1.7743 + i 71.528
0.13360
1.2865 + i 31.708
0.01551
0.8646 + i 76.363
-0.26841
2.3076 + i 38.490
-0.09364
1.3285 + i 78.662
0.32082
1.3828 + i 42.291
0.15913
1.2036 + i 83.669
-0.23365
0.9647 + i 48.847
-0.15736
2.3940 + i 85.802
-0.02503
2.1017 + i 52.432
0.11157
0.8641 + i 88.178
0.30930
1.8960 + i 57.135
-0.15093
1.3041 + i 93.086
-0.36793
0.8487 + i 60.141
0.25713
0.7806 + i 95.293
0.17932
1.2073 + i 65.920
-0.25909
1.7984 + i 98.827
0.04971
1.8330 + i 68.611
0.07686
Table C.2.: Zeros s00 of the derivative ζ 0 and phase of ζ at these points. The absolute value of
the phase is smaller than 0.37 < 0.12 π for s00 with imaginary part up to τ = 100.
104
C.5. Zeros of the derivative ζ 0
s00
s00
s00
s00
1.7671 + i 101.715
1.7758 + i 161.022
1.2142 + i 212.197
0.9150 + i 259.116
1.1606 + i 104.503
1.5504 + i 162.666
0.8772 + i 213.967
1.2240 + i 263.814
1.0925 + i 106.561
1.2010 + i 166.088
1.0786 + i 215.738
0.7917 + i 266.015
0.6356 + i 111.431
1.4141 + i 167.894
1.0282 + i 219.499
0.9555 + i 267.419
1.9060 + i 113.631
0.6455 + i 169.538
0.6210 + i 221.083
1.7119 + i 269.937
1.4473 + i 115.583
0.9288 + i 173.927
2.5097 + i 223.403
1.4125 + i 271.067
1.6571 + i 118.908
1.3442 + i 175.667
0.7463 + i 224.514
1.3726 + i 273.576
1.0226 + i 121.999
1.8222 + i 177.609
1.3869 + i 227.605
0.6955 + i 275.993
0.8478 + i 123.715
1.1700 + i 179.331
0.9281 + i 243.589
1.4412 + i 277.527
1.3792 + i 127.965
1.4466 + i 182.222
1.2864 + i 229.739
0.7509 + i 278.819
1.0570 + i 130.322
0.6160 + i 185.215
0.6305 + i 231.618
0.6408 + i 282.802
2.3285 + i 132.486
1.0499 + i 186.713
1.0561 + i 233.285
1.3150 + i 284.268
0.8563 + i 134.194
1.3982 + i 189.240
0.9255 + i 237.016
0.9281 + i 243.589
1.0433 + i 138.659
0.7739 + i 192.517
1.6714 + i 238.512
2.0256 + i 285.766
0.9438 + i 140.470
2.3054 + i 194.119
1.2642 + i 240.308
0.9911 + i 287.326
1.3056 + i 142.650
1.2421 + i 195.912
1.6496 + i 242.120
1.2213 + i 289.253
1.0188 + i 146.631
0.8109 + i 197.546
0.7282 + i 247.541
1.1848 + i 292.170
2.4238 + i 147.875
0.8654 + i 201.743
1.1059 + i 248.920
0.9719 + i 294.038
0.6629 + i 150.486
1.3098 + i 203.282
1.0648 + i 250.524
0.5921 + i 295.290
1.2660 + i 152.613
0.8653 + i 204.888
1.3697 + i 252.989
1.3321 + i 297.979
0.9670 + i 156.633
1.4953 + i 208.247
0.8258 + i 255.813
0.8634 + i 158.283
1.7404 + i 209.544
2.3571 + i 256.901
Table C.3.: Non-trivial zeros of ζ 0 with imaginary part 100 < τ < 300.
105
D. Phase space representations
From the variety of possible functions describing a quantum mechanical state in phase
space [20, 21, 28, 60–63], we concentrate on the function
Z∞
i
1
ξ
ξ
dξ e − ~ pξ hx + | ρ̂ |x − i
Wρ̂ (x, p) ≡
(D.1)
2π~
2
2
−∞
introduced by Wigner in 1932, which he had ‘chosen from all possible expressions, because it seems to be the simplest’ [18]. It allows a quasi-probabilistic interpretation of
the density operator ρ̂ and can be reconstructed from experiment, e.g. [64–66].
However, this definition can be generalized to non-Hermitian operators and even to
coupled systems leading to Moyal functions and Wigner matrices, respectively. We give
in this appendix a short overview over the properties of the different representations and
finish with a detailed description of the Wigner and Moyal functions of the Fock states
which are the basis of the states defined in this work.
D.1. Properties of the Wigner function
A special property of the Wigner function (D.1) is that it allows to calculate the probability distribution in one of the conjugate variables x and p by integrating over the other,
that is
Z∞
hx|ρ̂|xi = dp Wρ̂ (x, p)
(D.2)
−∞
for the position distribution and
Z∞
hp|ρ̂|pi = dx Wρ̂ (x, p)
(D.3)
−∞
for the momentum distribution of the operator ρ̂. Hence, the trace of this operator
Z∞ Z∞
Tr ρ̂ = dx dp Wρ̂ (x, p)
(D.4)
−∞
−∞
is given by integration over the whole phase space, as easily follows from eq. (D.2) as
well as from eq. (D.3). The overlap
Z∞ Z∞
Tr (ρ̂1 ρ̂2 ) = 2π~ dx dp Wρ̂1 (x, p) Wρ̂2 (x, p)
(D.5)
−∞
−∞
107
D. Phase space representations
of two operators ρ̂1 and ρ̂2 can be calculated from the product of their Wigner representations Wρ̂i , i = 1, 2. Obviously, the Wigner function is not a classical probability
distribution. Indeed, if two quantum states are orthogonal to each other the left hand
side of eq. (D.5) becomes zero. But positive definite Wigner functions on the right hand
side would only allow strictly positive outcomes. Hence, the Wigner functions must
become negative in some parts of the phase space. Therefore, they are called quasi
probability functions to emphasize their non-classical behavior.
The Wigner function is normally used to describe a quantum state in phase space,
that is the density operator ρ̂ is Hermitian, ρ̂ = ρ̂† , and normalized to unity, Tr ρ̂ = 1.
In this case, the Wigner function (D.1) becomes real and normalized to unity, eq. (D.4).
However, equations (D.2) – (D.5) hold for any ρ̂ since their proofs do not involve the
properties of the operator (see for example [28]). We will see in the next section that
it is useful to investigate Wigner functions of non-Hermitian operators, which are called
Moyal functions.
D.2. Moyal functions
We define the Moyal function [20, 21]
1
W|ψihφ| (x, p) ≡
2π~
Z∞
i
ξ
ξ
dξ e − ~ pξ hx + |ψihφ|x − i
2
2
(D.6)
−∞
as the generalization of the Wigner function (D.1) for two arbitrary states |ψi and |φi.
Hence, the Moyal function is complex, W|φihψ| = W|ψihφ| and reduces for |φi = |ψi to the
Wigner function W|ψihψ| ≡ W|ψi of the state |ψi.
As mentioned already in the previous section, eqs. (D.2) and (D.5) hold for any
operator ρ̂. Thus, we find for ρ̂ ≡ |ψihφ| that the normalization (D.4) simplifies to
Z∞ Z∞
dx dp W|ψihφ| = hφ|ψi ,
−∞
−∞
the scalar product of the two states – called ‘self-orthogonality’ in [20] – while the overlap,
eq. (D.5), is given by the expression
Z∞ Z∞
2π~ dx dp W|ψihα| W|γihφ| = Tr (|ψihα| |γihφ|) = hφ|ψihα|γi .
−∞
(D.7)
−∞
Since we have not used any information about the states, these equations are even valid
for unnormalized states. Needless to say, in the case of pure density operators, that is
for |φi = |ψi and |γi = |αi, the formulae reduce to the familiar ones of the Wigner
functions.
Until now, the density operator describes only one system, for example a harmonic
oscillator, but not a combination of two interacting systems. But with the help of
108
D.3. Wigner matrix
definition (D.1), we can even generalize the formalism to such systems, needed to describe
the states derived in this thesis.
D.3. Wigner matrix
Following the example of Wallentowitz [19], we define the Wigner matrix for the density
operator of two interacting systems A and B. Therefore, we use definition (D.1) on the
whole density operator with the modification that position and momentum apply only
to system B. Lets consider the density operator
XX
%̂AB ≡
%ab,a0 b0 |a, biha0 , b0 |
(D.8)
a,a0 b,b0
expanded in the orthonormal basis {|ai} and {|bi} of the systems A and B, respectively.
Definition (D.1) now leads to a Wigner matrix
X
Waa0 |aiha0 |
Ŵ%̂AB ≡
(D.9)
a,a0
with the Wigner matrix elements
Waa0 ≡
X
%ab,a0 b0 W|bihb0 | ,
(D.10)
b,b0
where the Moyal functions W|bihb0 | are given by eq. (D.6). It contains all information
of the motion of system B depending on system A’s degrees of freedom. Tracing over
system A leads to the Wigner function of the reduced density operator ρ̂B ≡ TrA %̂AB and
therefore a loss of all information about system A. Indeed, the Wigner function of the
reduced system ρ̂B is given by
X
Wρ̂B = TrA Ŵ%̂AB =
Waa ,
(D.11)
a
which only contains the diagonal matrix elements of the Wigner operator (D.9). This
representation clearly shows that information about the entanglement between the two
systems in %̂AB is contained in the off-diagonal matrix elements of the Wigner matrix.
Since the density operator of the two systems %̂AB is Hermitian, the Wigner matrix
is Hermitian, too, and therefore Waa0 = Wa0a . Moreover, the normalization TrAB %̂AB =
TrA ρ̂A = TrB ρ̂B = 1 of the density operator and the fact that TrA Ŵ%̂AB = Wρ̂B, eq. (D.11),
leads with eq. (D.4) to the normalization condition
Z∞ Z∞
Z∞ Z∞ X
Waa = 1
dx dp TrA Ŵ%̂AB = dx dp
−∞
−∞
−∞
−∞
a
for the Wigner matrix.
109
D. Phase space representations
Furthermore, we find the product of two density operators %̂AB and σ̂AB from
Z∞ Z∞
TrAB (%̂AB σ̂AB) = 2π~ dx dp TrA Ŵ%̂AB Ŵσ̂AB
−∞
Z∞
= 2π~
−∞
−∞
Z∞
dx
−∞
dp
X
Waa0 Ω aa0 .
(D.12)
a,a0
Here, Waa0 and Ωaa0 denote the matrix elements of the Wigner operators Ŵ%̂AB and Ŵσ̂AB,
defined by eq. (D.10), of the density operators %̂AB and σ̂AB , respectively. In the last step,
we made use of definition (D.9) to evaluate the trace. We now show the accuracy of this
equation.
Proof. The left hand side of eq. (D.12) reads with definition (D.8)


X X

TrAB (%̂AB σ̂AB) = TrAB 
%ab,a0 b0 %αβ,α0 β 0 |a, biha0 , b0 |α, βihα0 , β 0 |


a,a0 b,b0
α,α0 β,β 0
=
XX
a,a0
%ab,a0 b0 %a0 b0 ,ab .
(D.13)
b,b0
On the right, we arrive with eq. (D.10) at
X
a,a0
Z∞ Z∞
Z∞ Z∞
XX
%ab,a0 b0 % aβ,a0 β 0 2π~ dx dp W|bihb0 | W|βihβ 0 | .
2π~ dx dp Waa0 Ω aa0 =
−∞
a,a0 b,b0
β,β 0
−∞
−∞
−∞
Recalling that W|βihβ 0 | = W|β 0 ihβ| and that the Moyal functions fulfill eq. (D.7), this
expression becomes
XX
XX
%ab,a0 b0 %̄aβ,a0 β 0 δβ 0 b0 δβb =
%ab,a0 b0 %̄ab,a0 b0 .
(D.14)
a,a0 b,b0
β,β 0
a,a0 b,b0
Since %̂AB is Hermitian, that is %̄ab,a0 b0 = %a0 b0 ,ab , eq. (D.14) is equal to eq. (D.13).
Needless to say, tracing over system A after multiplication of the Wigner operators,
eq. (D.12), is not the same as if we multiply the Wigner functions of the reduced systems,
eq. (D.11), since this product would only consist of the diagonal matrix elements of the
Wigner matrices.
D.4. Fock states
Since in this work all states are given as expansions in the Fock basis, we now have a
closer look on their phase space distributions [28].
110
D.4. Fock states
Figure D.1.: Since the Wigner functions of the Fock states are rotational symmetric to the origin,
we have cut the pictures at p = 0 to emphasize the radial structure. The Wigner functions
W|ni , eq. (D.20), with even n have a maximum at the origin while the ones with odd n have a
minimum, each followed by n zeros in radial direction. The height of the wells decreases due to
the exponential envelope.
D.4.1. Wigner and Moyal functions
With the help of the position representation
r
2
κ
− κ2 x2
√
H
(κx)
e
hx|ni =
n
2 n n! π
(D.15)
of thepFock states |ni, where Hn denote the Hermite polynomials and the inverse length
κ ≡ (M ω)/~ contains the mass M and frequency ω of the harmonic oscillator, and
the substitution
ip
1
κx +
α≡ √
(D.16)
~κ
2
we get from eq. (D.6) the Moyal functions
W|nihm|
e −2|α|
=
π~
2
(
wnm (α) ᾱn−m
wmn (α)
The real function
wnm (α) ≡ (−1)m
r
αm−n
for n ≥ m
for n < m
.
m! n−m n−m
2
Lm
4|α|2
n!
(D.17)
(D.18)
contains the generalized Laguerre polynomials [50]
Ln−m
(x)
m
=
m X
k=0
n
m−k
(−x)k
.
k!
(D.19)
Hence, the shape of the phase space functions is governed by two contributions:
(i) the envelope exp{−2|α|2 } and
(ii) the real function wnm given by eq. (D.18).
111
D. Phase space representations
Figure D.2.: The real parts of the Moyal functions of the Fock states, eq. (D.21), are symmetric
in p if n − m is even, or antisymmetric for odd n − m. The star-shaped patterns are caused by
the real part of the complex factor exp −i(n − m)β, the zero circles are determined by the shape
of wnm . The imaginary parts of the Moyal functions look like the real ones but rotated by −π/2
n−m .
Since wnm only depends on |α|, it is rotational symmetric around the origin of phase
space. At the origin, it is positive for even m and negative for odd m, determined only
by the term (−1)m since the generalized Laguerre polynomials, eq. (D.19), are always
positive there. Moreover, the Laguerre polynomials have m different zeros for positive
values x which create circles around the origin in phase space. Needless to say, the
increase of the Laguerre polynomial to (±) infinity for x → ∞ is in W|nihm| suppressed
by the envelope.
Based on the previous observations about the function wnm , we see that the Wigner
functions
2
e −2|α|
W|ni (α) =
wnn (α)
(D.20)
π~
are symmetric around the origin, where – depending on n – we have a maximum (n
even) or minimum (n odd). Around that, we find the ridges and valleys caused by the
minima and maxima of Ln and shaped by the envelope, as we see in fig. D.1.
The zero circles around the origin caused by the zeros of the Laguerre polynomials
also occur in the Moyal functions. However, when we rewrite eq. (D.17) for n > m
2
W|nihm|
e −2|α|
=
wnm (α) |α|n−m e −i(n−m)β(α)
π~
(D.21)
we find an additional zero at the origin, due to the factor |α|n−m . Moreover, the phase
factor exp{−i(n − m)β}, depending on the angle β ≡ arg α, scales the values of the real
or imaginary part of W|nihm| with the factor cos[(n − m)β] or − sin[(n − m)β]. This
π/2
π
creates radial zero lines at β = (2k + 1) (n−m)
and β = k n−m
, k ∈ N, respectively.
Hence, Re W|nihm| is symmetric in p while Im W|nihm| is antisymmetric.
The star-shaped patterns of the real part of the Moyal functions are shown in fig. D.2.
The shapes of the imaginary part are equal to the real part patterns rotated by −π/2
n−m .
112
D.4. Fock states
N =2
N =3
N = 20
0.3
0.2
0.1
-4
-2
2
4
-4
2
-2
4
-8
-4
4
8
x
Figure D.3.: Behavior along the x-axis of the absolute value of the Wigner function |W|N i |,
eq. (D.20), compared to the sum SN −1 , eq. (D.22), for different N , indicated in red and blue,
respectively. The shaded areas mark the regions where W|N i is negative. Since |W|N i | in these
regions is always smaller than SN −1 , the sum SN is positive in the whole phase space.
D.4.2. Sum over Wigner functions
Due to the normalization of the Wigner function, the absolute value of W|1i around the
origin is smaller than the Gauß shape of W|0i . This holds for the whole region where
W|1i < 0. Thus, the sum of the two Wigner functions, W|0i + W|1i , is positive in the
whole phase space. When we continue to sum up the Wigner functions W|ni , we find
the same result:
N
X
SN ≡
W|ni ≥ 0 .
(D.22)
n=0
Indeed, when we examine the absolute value of the Wigner function W|N i , given by the
red line in fig. D.3, we see that it is smaller than the sum SN −1 (blue line) in the region
where W|N i is negative (indicated by the shaded areas). Hence, SN ≡ SN −1 + W|N i is
positive in the whole phase space. This is, of course, no mathematical proof, but there
are hints which strongly suggest that eq. (D.22) holds for all N :
(i) The value of the Wigner functions at the origin is +1 for even n and −1 for odd
leading to S2N = 1 and S2N +1 = 0.
(ii) The peak around the origin of |W|n+1i | is narrower than the peak of |W|ni |, which
leaves a positive contribution from each summand W|2ni + W|2n+1i . Thus, SN is
always positive around the origin.
(iii) The higher order peaks of |W|N i | decrease exponentially in height for increasing
|x| and become less important in comparison to the relatively large value of SN −1
as the pictures in fig. D.3 confirm.
(iv) The last peak of W|N i for x → ∞ is always positive.
From the positivity of the sum SN follows that the contribution of the sum
N
X
n=0
W|ni
(n + 1)σ
in the Wigner function of the truncated Riemann state, eq. (5.6), is positive, too.
113
E. Entanglement
In this appendix, we will show different ways to distinguish, if a pure state of the form
X
ψne |n, ei + ψng |n, gi
(E.1)
|ΨiAB ≡
n
is entangled or not. Since we have used this compact notation for the state |Ψi to
describe the Riemann zeta function, we return to it as often as possible. But for the
sake of simplicity, we switch notation if it is convenient.
E.1. Quick check
A quantum state is entangled, if we cannot factorize it in a product of two states of the
independent systems. The pure state
|Ψi ≡ N c1 |α1 i|ei + c2 |α2 i|gi ,
(E.2)
consisting of the atomic states |ei and |gi (of system A) and the normalized states |α1 i
and |α2 i of system B, can only be written as a product state if |α1 i and |α2 i are linearly
dependent, that is |α2 i = e iϕ |α1 i. Hence, the overlap fulfills
|hα1 |α2 i|2 ≤ 1
(E.3)
with equality only given for product states.
The information how strong |Ψi is entangled can be extracted from so-called entanglement measures. For an introduction to entanglement measures see for example [67].
Since we are only interested in the question whether a state is entangled or not, we leave
the question of how much for further investigations.
E.2. Schmidt decomposition
It can be shown [68] that every composite pure state
|ΨiAB ≡
dA X
dB
X
α
β
cαβ |αi|βi
can be written in its Schmidt decomposition
|ΨiAB ≡
k p
X
λi |ai i|bi i ,
(E.4)
i=1
115
E. Entanglement
where λi are the non-zero eigenvalues of the reduced density matrix ρ̂A. The states
|ai i are the corresponding eigenstates and |bi i are orthonormal states, too. From the
properties of the density matrix follows
λi > 0
and
k
X
λi = 1 .
i=1
Moreover, we can show from the Schmidt decomposition, eq. (E.4), that the reduced
density matrix ρ̂B has the same eigenvalues λi as ρ̂A.
Another consequence of the Schmidt decomposition is that if and only if |ΨiAB and
thus ρ̂AB is not entangled, the Schmidt decomposition has only one term with eigenvalue
λ1 = 1 and the reduced density operators are pure. Hence, the Schmidt decomposition
is a powerful tool to distinguish entangled states from product states.
Now, we use eq. (E.1) to determine the eigenvalues of the reduced density matrix ρ̂A,
with atom states belonging to system A and photon states to system B. Tracing over
the photon states yields
T
ρ̂A ≡ TrB ρ̂AB = |ei, |gi A he|, hg|
with the Hermitian matrix
 P

A ≡  Pn
n
|ψne |2
ψne ψng

ψne ψng

n
≡
P
2
|ψng |
P
n
a
b
b̄ 1 − a
!
.
Here, Tr A = 1 results from the normalization of |Ψi. Moreover, Hermiticity implies
that the eigenvalues
r
1
1
λ1/2 ≡ ±
+ |b|2 − a(1 − a)
(E.5)
2
4
of the matrix A have to be real, leading to the inequality
a(1 − a) ≤ |b|2 +
1
.
4
We note that in this expression equivalence produces degenerated eigenvalues λ1/2 = 1/2.
However, as a density operator ρ̂A is positive semi-definite. Thus, the non-negativity
of the eigenvalues, eq. (E.5), yields
|b|2 ≤ a(1 − a) .
(E.6)
In the case of equality, we get λ2 = 0 and the Schmidt decomposition consists of a single
non-zero term. Therefore, ρ̂A describes a pure state and ρ̂AB is disentangled. Needless to
say, eq. (E.6) is equivalent to the condition Det A ≥ 0.
116
E.3. Wigner representation
In terms of the probability amplitudes ψne and ψng , eq. (E.6) reads
2
X
X
X
|ψmg |2
ψne ψng ≤
|ψne |2
n
n
(E.7)
m
which we use in section E.4 and E.5 to investigate the entanglement of the RiemannSiegel and Berry-Keating reference state. By using the correspondence
X
ψne/ng |ni ≡ N c1/2 |α1/2 i ,
n
one can show that eq. (E.7) is equivalent to eq. (E.3).
E.3. Wigner representation
For the sake of completeness, we answer the question how entanglement effects the
Wigner matrix of the state |Ψi, eq. (E.2). Therefore, we recombine the Wigner matrix
elements of Ŵ|Ψi in
W≡
Wee Weg
Weg Wgg
!
= N2
|c1 |2 W|α1 i
c̄1 c2 W|α2 ihα1 |
c1 c̄2 W|α1 ihα2 |
|c2 |2 W|α2 i
!
and integrate the trace of this matrix over the whole phase space. The result is always
(2π~)−1 , since the Wigner functions W|α1 i and W|α2 i are normalized to unity.
However, with the help of eqs. (D.5) and (D.7), we find that integration of the determinant of the matrix yields
Z∞ Z∞
2
2
2
2
2π~ dx dp Det W = N |c1 | |c2 | |hα1 |α2 i| − 1 ≤ 0 .
−∞
(E.8)
−∞
As in eq. (E.3), equality indicates a product state and negative values represent entanglement. Moreover, a direct evaluation of Det W for a product state results in a
vanishing determinant, which can never occur for an entangled state since eq. (E.8)
must be fulfilled. Thus, we get the equivalent formulations
Det W = 0
⇔
|α2 i = e iϕ |α1 i
for the state |Ψi.
117
E. Entanglement
E.4. Riemann-Siegel reference state
We note in chapter 6 that the Riemann-Siegel reference state is only entangled on the
critical line which is equivalent to the postulate that equality in eq. (6.10) is only given
for n0 ≥ 2 if σ = 1/2. We now proof this statement.
Proof. Lets substitute a ≡ σ − 1, b ≡ 1 − 2σ and xn ≡ n + 1 in eq. (6.10), that is
nX
0 −1
xan
n=0
!2
≤
nX
0 −1
xa+b
xa−b
n
m .
(E.9)
n,m=0
Obviously, both sides are equal if n0 = 1.
For n0 = 2 we get
a+b
a−b
a−b
≤ xa+b
+
x
x
+
x
0
1
0
1
" −b #
b
x
x0
a
a
0
2a
2a
x2a
≤ x2a
+
0 + x1 + 2 x0 x1
0 + x1 + x0 x1
x1
x1
x0
.
1 ≤ cosh b ln
x1
⇔
⇔
xa0 + xa1
2
The right hand side is only equal to one if the argument of cosh is zero, which is the
case if b = 0 and therefore σ = 1/2. Otherwise, that is beside the critical line, cosh is
larger than one.
Lets now assume that equality in eq. (E.9) holds for a definite n0 > 2 only if σ = 1/2.
Then follows for n0 + 1
nX
0 −1
xan
+
xan0
n=0
⇔
⇔
nX
0 −1
xan
!2
xan
!2
n=0
nX
0 −1
n=0
+
2 xan0
nX
0 −1
!2
xan
n=0
+
2 xan0
nX
0 −1
n=0
xan
≤
≤
≤
nX
0 −1
xa+b
n
+
xa+b
n0
n=0
nX
0 −1
a−b
xa+b
n xm
+
n,m=0
nX
0 −1
n,m=0
!
xan0
nX
0 −1
xa−b
m
m=0
nX
0 −1
xan
n=0
a−b
xa+b
n xm
+
2 xan0
nX
0 −1
n=0
+
"
xa−b
n0
xn0
xn
xan cosh
b
!
+
b ln
xn0
xn
xn0
xn
−b #
.
Again, the cosh-term is larger than 1 for σ 6= 1/2 which makes the second term on the
right larger than the second one on the left. Since the first terms already fulfill eq. (E.9),
the statement that equality is only given for σ = 1/2 also holds for n0 + 1.
118
E.5. Berry-Keating reference state
E.5. Berry-Keating reference state
We show now that – in contrast to the Riemann-Siegel reference state – the BerryKeating reference state eq. (7.3) is always entangled. For this task, we use again eq. (E.7)
which yields for the probability amplitudes
en (σ, τ ) N
en (σ, τ )
φne ≡ N
B
B
and
the inequality
E (n, τ )
(n + 1)σ/2
en (σ, τ ) N
en (2 − 3σ, τ ) γ (σ, τ )
φng ≡ N
B
B
B
E(n, τ )
(n + 1)(2−3σ)/2
n −1
2
nX
B
B −1
X
E 2 (n, τ ) |E(n, τ )|2 |E(m, τ )|2
≤
.
(n + 1)1−σ (n + 1)σ (m + 1)2−3σ
n=0
(E.10)
n,m=0
e 4 (σ, τ ) N
e 2 (σ, τ ) N
e 2 (2−3σ, τ ) |γ (σ, τ )|2
Here, we have already neglected the factors N
nB
nB
nB
B
on both sides.
Applying the triangle inequality on the left hand side, that is
n −1
2 n −1
B
B
X
X
E 2 (n, τ ) |E(n, τ )|4
≤
,
(n + 1)1−σ (n + 1)2−2σ
n=0
n=0
and splitting the double sum on the right of eq. (E.10) into
nX
B −1
n,m=0
nX
nX
B −1
B −1
|E(n, τ )|4
|E(n, τ )|2 |E(m, τ )|2
|E(n, τ )|2 |E(m, τ )|2
=
+
(n + 1)σ (m + 1)2−3σ
(n + 1)2−2σ
(n + 1)σ (m + 1)2−3σ
n=0
n,m=0
n6=m
immediately reveals that the right hand side must always be larger than the left due to
the terms with n 6= m. Hence, the Berry-Keating reference state is entangled for all σ
and τ .
119
F. Normalization and proportionality factors
We give here a short investigation of the normalization and proportionality factors appearing throughout this work.
F.1. Truncated Riemann state
The normalization, eq. (5.5),
Nν (σ) ≡
ν−1
X
n=0
1
(n + 1)σ
!−1/2
(F.1)
of the truncated Riemann states |σ, τ iν is involved in the description of the zeta function
by the truncated alternating sum and the Riemann-Siegel formula in chapter 5 and 6.
Moreover, the limit N∞ (σ) ≡ N (σ) = ζ −1/2 (σ) causes the restriction to σ > 1 of the
Riemann states |σ, τ i in chapter 4.
In fig. F.1, we show the behavior of Nν2 for different values ν in dependence on σ.
The curves start at Nν (0) = 1/ν and increase until they reach the limit Nν (∞) = 1.
Hence, the curve of Nν with larger summation limit is always below the ones with smaller
ν. The lowest curve, indicated in black, displays the normalization N of the Riemann
states which is only defined outside the critical strip. The dashed and the dotted gray
line mark the values for σ = 1/2 and for σ = 1, respectively.
Nν2
1.0
0.8
0.6
0.4
0.2
2
4
6
8
10
σ
Figure F.1.: The curves of the normalization factor Nν2 , eq. (F.1), given for the values ν = 1
(orange), ν = 2 (blue), ν = 3 (purple) and ν = 20 (red), approach the limit one for large σ. Since
they start at Nν2 (0) = 1/ν, the curves with larger ν are always below the ones with smaller ν.
The lowest curve is given by the normalization N (black) which is only defined for σ > 1. The
dashed and the dotted gray line mark σ = 1/2 and σ = 1, respectively.
121
F. Normalization and proportionality factors
F.2. The function E(n, τ )
For a better understanding of the Berry-Keating formula, eq. (7.1), and the Berry states
defined in chapter 7, we give here a short analysis of the behavior of the function
#
"r
1
τ
n+1
E(n, τ ) ≡ Erfc
(F.2)
ln p τ
2
1−i
2π
which involves the complementary error function Erfc, eq. (B.8) [56].
Since the behavior of E is the same for all τ , we show in fig. F.2 only a sketch of its
absolute value and argument for a fixed value τ and the continuous p
variable n with dots
marking integer n. The vertical dashed orange line indicates n = τ /(2π) − 1, where
E = 1/2. Thus, the first dot on the right of this line marks n0 . Both pictures show that
for almost all n ≤ n0 the absolute value and the argument of E is constant, that is equal
to one or zero, respectively. However, for n > n0 the absolute value is zero while the
argument rotates clockwise in the complex plane. Only the values near the orange line
deviate from this behavior.
The property |E| = 0 for n > n0 results from the behavior of the complementary error
function and was used by Berry and Keating [31] to cut the sum in ZBK, eq. (1.12), at
nB ≥ n0 . Moreover, we show in section 7.3 that the probability amplitudes in the Berry
state |φBinB , defined by eq. (7.4), are almost the same as for the initial truncated Riemann
state |σ, 0inB , eq. (5.4), due to the fact that E = 1 for small values of n. The largest
deviations between |φBinB and |σ, 0inB occur at the discontinuities τ = 2π n20 of the
Riemann-Siegel formula, eq. (1.9), and fade between the discontinuities for increasing τ .
This behavior becomes clear when we consider that between the jumps the dots marking
the integers of n are shifted to the left on the curves shown in fig. F.2.
Figure F.2.: Sketch of the function E, defined by eq. (F.2), for a fixed time τ in dependence on
a continuous
variable n with
p
p dots marking integers. The dashed vertical orange line indicates
n = τ /(2π) − 1, where E( τ /(2π) − 1, τ ) = 1/2. The absolute value |E|, depicted on the left,
is for almost all n < n0 equal to one and drops after a slight increase in the vicinity of n0 − 1
to zero. In contrast, the argument of E, pictured on the right, is at first zero, becomes slightly
positive around n0 − 1 and then rotates clockwise in the complex plane for n on the right of the
orange line.
122
F.3. Berry state
F.3. Berry state
The norm of the Berry state |φB(σ, τ )i, eq. (7.4), reads
en (σ, τ ) ≡
N
B
"n −1
#−1/2
B
X
|E(n, τ )|2
n=0
(n + 1)σ
.
(F.3)
Due to the τ -dependence of the function E, eq. (F.2), we get different normalization
factors for different values of τ even if the summation limit nB ≥ n0 (τ ) is the same. The
en2 for nB = n0 (τ ) chosen for different values of τ . Like
left picture in fig. F.3 shows N
B
en2 increases to the limit one. However, it starts
the curves of Nν2 depicted in fig. F.1, N
B
at
"n −1
#−1
B
X
2
2
en (0, τ ) =
N
|E(n, τ )|
B
n=0
e2 − N2
which is larger than NnB (0) = 1/nB if nB > 1. Moreover, the difference N
nB
nB
between the normalizations of |φB(σ, τ )inB and |σ, τ inB are distinct at the jumps τ = 2πn20
of the Riemann-Siegel formula, eq. (1.9), and tend to zero for increasing τ until the next
jump occurs, as the right pictures in fig. F.3 confirm. Of course, the difference fades also
for increasing summation limit nB .
e2
N
n0
en2 − Nn2
N
0
0
1.0
0.3
0.2
0.8
0.1
0.6
0.003
0.4
0.002
0.2
0.001
2
4
6
8
10
2
4
6
8
10
σ
en2 , eq. (F.3), for 1 < nB = n0 (τ ) at
Figure F.3.: The curves depicting the normalization factor N
0
different values of τ approach the limit one for large σ, just like the norm Nν2 shown in fig. F.1.
However, due to the τ -dependence of E, the curves are different for different values of τ and they
start at larger values compared with the ones of Nn20 . The right pictures show the difference
e 2 − N 2 between the normalizations. It is largest at the jumps τ = 2πn2 (solid lines) of the
N
n0
n0
0
Riemann-Siegel formula and tends to zero before the next jump at 2π(n0 + 1)2 . As expected,
the difference also becomes smaller for increasing summation limit. The dashed and the dotted
gray line mark σ = 1/2 and σ = 1, respectively. The parameters are chosen as follows:
n0 = 2 (blue): τ = 8π ∼
= 25.1 (solid), τ = 30 (dashed), τ = 56 (dotted);
n0 = 3 (purple): τ = 18π ∼
= 56.5 (solid), τ = 75 (dashed), τ = 100 (dotted);
n0 = 20 (red): τ = 800π ∼
= 2513.3 (solid), τ = 2600 (dashed), τ = 2770 (dotted).
123
F. Normalization and proportionality factors
F.4. Truncated alternating sum
In section 5.3.1, the proportionality factor, eq. (5.10),
)
M(ν,N
(σ, τ ) ≡ NN (σ) Nν (σ) (21−σ−iτ − 1)
a
(F.4)
appears, connecting the overlap C, eq. (5.8), with the truncated alternating sum ζN ,
eq. (5.7). For the sake of simplicity, we have chosen there ν = N in all pictures. Hence,
(N,N )
(N )
we investigate now only the behavior of Ma
≡ Ma .
(N )
The factor Ma displays a sinusoidal dependence on τ , as fig. F.4 shows. Moreover,
its absolute value as well as the absolute value of its real and imaginary part decrease
(N )
for increasing truncation parameter N or decreasing value σ. Hence, Ma yields very
small values of the overlap C if we improve the approximation ζN of the Riemann zeta
function by increasing the truncation parameter N .
(N )
|Ma (σ, τ )|
0.3
0.2
0.1
2
4
6
8
10
12
n
o
(N )
Re Ma (σ, τ )
2
4
τ
n
o
(N )
Im Ma (σ, τ )
0.2
6
8
10
12
τ
0.1
-0.1
2
4
6
8
10
12
τ
-0.2
-0.1
-0.3
-0.2
(N )
Figure F.4.: The dependence on τ of the proportionality factor Ma (σ, τ ), defined by eq. (F.4)
for ν = N , displays a sinusoidal shape for the absolute value as well as for the real and imaginary
part. As expected, the values for N = 100, indicated by the red curves, are smaller than for
(N )
N = 50 (black) and N = 30 (blue). The solid and dashed lines indicate Ma for σ = 2/3 and
σ = 1/2, respectively, to emphasize that the curves of smaller σ are closer to the τ -axis.
124
F.5. Riemann-Siegel reference state
F.5. Riemann-Siegel reference state
Like Nν2 (see section F.1), the square of the normalization, eq. (6.8),
1
Nn0 (σ, τ ) ≡ q
1 + |γ(σ, τ )|2
with γ(σ, τ ) ≡
Nn0 (σ)
χ(σ + iτ )
Nn0 (2 − 3σ)
(F.5)
of the Riemann-Siegel reference state |ΦRSi increases with increasing σ from values smaller
than 1/2 to the limiting value 1, as the upper pictures of fig. F.5 shows for three different
τ for each value n0 . However, the curves of Nn20 cross each other on the critical line where
Nn20 = 1/2.
In the Wigner matrix Ŵ|ΦRSi , eq. (6.15), the function γ appears as additional factor
which inverts the behavior of Nn20 . Indeed, the absolute value |γ| Nn20 in the off-diagonal
matrix elements (middle row of fig. F.5) increases for σ < 1/2 before it tends to zero.
In contrast, the factor |γ|2 Nn20 in the diagonal matrix element connected to |gi (bottom
row) starts at values larger than 1/2 before dropping to zero. Due to the fact that
|γ| = 1 holds on the critical line independently of τ , the absolute values of the factors in
all matrix elements are the same for σ = 1/2.
F.6. Berry-Keating reference state
For the Berry-Keating reference state |ΦBKi the normalization, eq. (7.6), reads
1
en (σ, τ ) ≡ q
N
B
1 + |γB(σ, τ )|2
with
γB(σ, τ ) ≡
en (σ, τ )
N
B
en (2 − 3σ, τ )
N
B
χ(σ + iτ ) .
(F.6)
The structure of the formula is the same as for the norm Nn0 , eq. (F.5), of the Riemanneν of the Berry states, (F.3), are involved.
Siegel reference state. But now the norms N
However, for σ =
√ 1/2 these norms cancel each other and we get the same value
en (1/2, τ ) = 1/ 2 as for Nn (1/2). Moreover, N
e1 (σ, τ ) = N1 (σ, τ ) since the factor
N
0
B
−1
e
N1 = |E(0, τ )| is independent of σ and therefore γB = γ.
en2 for nB = n0 (τ ) shown in the first row of fig. F.6 look like
Although the curves of N
B
the ones for Nn0 depicted in fig. F.5, they are slightly different. The inset confirms that
en − Nn is largest at the at the jumps τ = 2πn2 and fades in the interval
the difference N
0
0
0
2
2πn0 ≤ τ < 2π(n0 + 1)2 and, of course, with increasing summation limits.
The bottom pictures in fig. F.6 reveal that the factor |γB| is more restricting than |γ|.
e 2 in the middle row do not exceed the value 1/2 and the ones in the
The curves of |γB| N
n0
en2 ≤ 1 for σ < 1/2. Again, all factors are the
bottom row are limited to 1/2 ≤ |γB|2 N
0
same on the critical line.
125
F. Normalization and proportionality factors
Figure F.5.: The factors appearing in the Wigner matrix elements of Ŵ|ΦRSi , eq. (6.11), differ
in their behavior. The left pictures show the dependence on σ up to 10 while the right ones
are magnifications of the critical strip. The first row depicts the factor Nn20 , eq. (F.5) in the
diagonal matrix element connected to the excited state, which increases for increasing σ from
values smaller than 1/2 to 1. However, the factor |γ| in the absolute value of the prefactor of
the off-diagonal matrix elements (second row) causes a moderate increase for σ < 1/2 before the
curves drop to zero. In contrast to that, |γ|2 in the prefactor of the diagonal matrix element
connected to |gi generates a decreasing curve. The dashed gray line indicates the critical line
σ = 1/2, where the prefactors are identical since |γ| = 1. The dotted gray line marks σ = 1. The
curves are depicted for the following values:
n0 = 1 (orange): τ = 2π ∼
= 6.3 (solid), τ = 10 (dashed), τ = 25 (dotted);
n0 = 2 (blue): τ = 8π ∼
25.1
(solid), τ = 30 (dashed), τ = 56 (dotted);
=
n0 = 20 (red): τ = 800π ∼
2513.3
(solid), τ = 2600 (dashed), τ = 2770 (dotted).
=
126
F.6. Berry-Keating reference state
e2
N
n0
1.0
0.5
0 10
0 05
2
4
6
8
10
σ
-0 05
1.0
02
2
4
6
8
10
σ
2
4
6
8
10
σ
-0 2
e2
|γB| N
n0
-0 4
0.5
1.0
02
en2
|γB|2 N
0
-0
-0
-0
-0
-1
2
4
6
8
0
0.5
2
4
6
8
10
0.2
0.4
0.6
0.8
1.0
σ
Figure F.6.: The factors in the Wigner matrix elements of Ŵ|ΦBKi , eq. (7.10), resemble in their
en2 , eq. (F.6), connected to
behavior the factors in Ŵ|ΦRSi , shown in fig. F.5: (i) The factor N
0
|ei (first row) increases to the limit 1. (ii) The absolute value of the one in the off-diagonal
matrix element (second row) first increases and then drops to zero. (iii) The one connected to
|gi (bottom row) only decreases and (iv) all curves cross each other on the critical line (dashed
gray line) at the value 1/2. However, the higher resolution of the critical strip shown on the
right reveals that |γB| changes the behavior of the factors especially for σ < 1/2: The factor in
the off-diagonal matrix elements (middle picture) does not exceed the value 1/2 and (bottom)
e 2 ≤ 1 for σ = 0, in contrast to the behavior of |γ| N 2 and |γ|2 N 2 depicted in fig. F.5.
|γB|2 N
n0
n0
n0
en2 − Nn2 , |γ | N
en2 − |γ| Nn2 and |γ |2 N
en2 − |γ|2 Nn2
In the left column, the inset pictures of N
B
B
0
0
0
0
0
0
emphasize the difference between the factors in Ŵ|ΦBKi and Ŵ|ΦRSi . The dotted gray line indicates
the border of the critical strip σ = 1 and the colors represent the following values:
n0 = 1 (orange): τ = 2π ∼
= 6.3 (solid), τ = 10 (dashed), τ = 25 (dotted);
n0 = 2 (blue): τ = 8π ∼
= 25.1 (solid), τ = 30 (dashed), τ = 56 (dotted);
n0 = 20 (red): τ = 800π ∼
= 2513.3 (solid), τ = 2600 (dashed), τ = 2770 (dotted).
127
G. Effective Hamiltonian
We give here a short derivation of eq. (8.7) for the effective Hamiltonian.
G.1. Time-dependent Hamiltonian
Since our calculations are carried out in the interaction picture, we have to deal with
a time-dependent Hamiltonian Ĥ(t). The solution of the time-dependent Schrödinger
equation
d
i~ |ψ(t)i = Ĥ(t)|ψ(t)i
dt
for this Hamiltonian is formally given by
|ψ(t)i = Û (t)|ψ(0)i ,
where the time-evolution operator



 i Zt
dt0 Ĥ(t0 ) 
Û (t) ≡ T̂ exp −

 ~

0
contains the time ordering operator
h
i
T̂ Ĥ(t1 ) Ĥ(t2 ) =
(
Ĥ(t1 ) Ĥ(t2 ) , t2 ≤ t1
Ĥ(t2 ) Ĥ(t1 ) , t1 < t2
.
One can show [28, 45] that
 t

tZν−1
Z
Zt
Zt
1
T̂  dt1 . . . dtν Ĥ(t1 ) · . . . · Ĥ(tν ) = dt1 . . . dtν Ĥ(t1 ) · . . . · Ĥ(tν ) .
ν!
0
0
0
0
Hence, the time-evolution operator is approximately
Û (t) ∼
= 1̂ −
i
~
Zt
0
dt0 Ĥ(t0 ) −
1
~2
Zt
0
dt0
Zt0
dt00 Ĥ(t0 )Ĥ(t00 )
(G.1)
0
up to second order in time. We will now use a special form of the Hamiltonian, following
from the transformation into the interaction picture, to calculate Û (t).
129
G. Effective Hamiltonian
G.2. Interaction picture and effective Hamiltonian
The transformation of a Hamiltonian Ĥ ≡ Ĥ0 + Ĥint with the free Hamiltonian Ĥ0 and
the interaction Hamiltonian Ĥint into the interaction picture is given by
i
i
I
Ĥint
≡ e ~ Ĥ0 t Ĥint e − ~ Ĥ0 t
and yields a Hamiltonian of the form
I
Ĥint
(t) ≡
X
Ĥj σ̂ e −iνj t + Ĥj† σ̂ † e iνj t
j
in the case of the Jaynes-Cummings-Paul model. Here, σ̂ ≡ |gihe| is the Pauli operator.
The first order term of the time-evolution operator, eq. (G.1), then contains the integral
#
"
Zt
†
X Ĥj σ̂
Ĥj σ̂ † iν t
−iνj t
0
0
e
−1 −
e j −1
I1 ≡ dt Ĥ(t ) = i
νj
νj
0
j
and the second order contribution is proportional to
I2 ≡
Zt
0
=i
dt0
Zt0
dt00 Ĥ(t0 )Ĥ(t00 )
0
t
XZ
k,j 0
dt0
"
#
Ĥk Ĥ † σ̂σ̂ † Ĥk† Ĥj σ̂ † σ̂ i(νk −νj )t0
0
0
0
j
e
− e iνk t −
e −i(νk −νj )t − e −iνk t
.
νj
νj
Here, we have used the fact that σ̂ 2 = 0. When we now carry out the integration over
t0 , we arrive at
i
X i h †
Ĥj Ĥj σ̂ † σ̂ − Ĥj Ĥj† σ̂σ̂ † t
νj
j
X 1 h †
i
†
†
iνj t
†
−iνj t
−
Ĥ
Ĥ
σ̂
σ̂
e
−
1
+
Ĥ
Ĥ
σ̂σ̂
e
−
1
j j
j j
νj2
j
"
!
X
e i(νk −νj )t − 1 e iνk t − 1
†
†
+
Ĥk Ĥj σ̂ σ̂
−
νj (νk − νj )
νk νj
j6=k
!#
e −i(νk −νj )t − 1 e −iνk t − 1
†
†
−
.
+ Ĥk Ĥj σ̂σ̂
νj (νk − νj )
νk νj
I2 =
The second and third contribution contain oscillatory terms and can be neglected since
they are of the order O(ν −2 ) and therefore much smaller than the first term (O(ν −1 )).
Thus, we only keep the contributions of I2 which are linear in time.
130
G.2. Interaction picture and effective Hamiltonian
Moreover, the first order term I1 can be neglected in comparison to I2 since its
contributions are constant, creating an overall phase, or oscillate much faster in time than
the linear term of I2 . Hence, the time-evolution operator, eq. (G.1), is approximately
given by
†
†
†
†
it X Ĥj Ĥj σ̂ σ̂ − Ĥj Ĥj σ̂σ̂
.
(G.2)
Û (t) ∼
1̂
−
=
~
~νj
j
Up to first order in time t, eq. (G.2) is equivalent to the time evolution produced by an
effective Hamiltonian
X Ĥj† Ĥj σ̂ † σ̂ − Ĥj Ĥj† σ̂σ̂ †
Ĥeff ≡
~νj
j
which transforms into
i
X 1 h †
Ĥj Ĥj − Ĥj Ĥj† 1̂ + Ĥj† Ĥj + Ĥj Ĥj† σ̂z
Ĥeff ≡
2~νj
j
when we use
1̂ ± σ̂z
=
2
(
|eihe| = σ̂ † σ̂
|gihg| = σ̂σ̂ † .
This formula is employed to calculate the effective Hamiltonian of the Jaynes-CummingsPaul model and to approximate the Riemann Hamiltonian ĤR in chapter 8.
131
H. Matrix elements
In chapter 8, we need the matrix elements hn + j|fˆ|ni to find an effective Hamiltonian
which approximates the Riemann Hamiltonian ĤR , eq. (3.2). We will now give the
calculation of these matrix elements for a polynomial operator fˆ ≡ f (x̂).
H.1. Calculation with â and â†
When we recall that the position operator
κ x̂ ≡ √
â + â†
2
for the harmonic oscillator is the sum of the annihilation operator â and the creation
operator ↠, we immediately see that |hn ± j|x̂µ |ni|2 is a polynomial in n of degree µ
(see tab. H.1 for µ = 0, . . . , 4). Therefore, we find for a polynomial operator
fˆ ≡ f (x̂) =
that
µX
max
fµ x̂µ
µ=0
2
max
2 µX
µ
ˆ
fµ hn ± j|x̂ |ni
hn ± j|f |ni = µ=0
is a polynomial of degree nµmax . A closer examination of |hn ± j|x̂µ |ni|2 shows that only
matrix elements with j ≤ µ survive, that is
2
2
(H.1)
hn ± j|fˆ|ni = |f0 + f1 g1 + f2 g2 + · · · + fµmax gµmax | ,
where
gν ≡
 ν
2
P



pk δj,2k ,


 k=0

ν+1


2
P



pek δj,2k−1 ,
ν even
ν odd .
k=1
Here, pν denotes a polynomial of order ν and peν represents the square root of a polynomial of order 2ν − 1. The Kronecker deltas δj,ν cause a splitting of the absolute square
in eq. (H.1) into
2
2
2
(H.2)
hn ± j|fˆ|ni = |f0 + f2 g2 + . . . | + |f1 g1 + f3 g3 + . . . |
133
H. Matrix elements
µ
hn − j| (â + ↠)µ |ni
hn + j| (â + ↠)µ |ni
0
1
2
3
4
δj,0
√
n δj,1
p
(2n + 1) δj,0 + n(n − 1) δj,2
p
3n3/2 δj,1 + n(n − 1)(n − 2) δj,3
(6n2 + 6n + 3) δj,0
p
+(4n − 2) n(n − 1) δj,2
p
+ n(n − 1)(n − 2)(n − 3) δj,4
√
n + 1 δj,1
p
(2n + 1) δj,0 + (n + 1)(n + 2) δj,2
p
3(n + 1)3/2 δj,1 + (n + 1)(n + 2)(n + 3) δj,3
(6n2 + 6n + 3) δj,0
p
+(4n + 6) (n + 1)(n + 2) δj,2
p
+ (n + 1)(n + 2)(n + 3)(n + 4) δj,4
Table H.1.: Matrix elements hn ± j| (â + ↠)µ |ni for n = 0, . . . , 4.
the absolute square of the terms with µ odd and µ even, respectively. Furthermore, this
splitting causes the first part of eq. (H.2) to be a polynomial of degree 2ν in n whereas
the second part, containing the odd coefficients, is a polynomial of order 2ν − 1.
H.2. General expression
Here, we derive a general formula for the matrix elements which can be used to numerically calculate them. Therefore, we use again the series expansion
f (x̂) ≡
∞
X
fµ x̂µ
µ=0
of the operator fˆ. The matrix elements are then given by
hn + j|f (x̂)|ni =
∞
X
µ=0
µ
fµ hn + j|x̂ |ni =
2n
p
κ
π n! 2j (n + j)!
∞
X
µ=0
fµ Ien,j,µ .
(H.3)
In the last step, we made use of the position representation of the Fock states, eq. (D.15),
to express the integral by
Ien,j,µ ≡
Z∞
2
dx xµ Hn+j (κx) Hn (κx) e −(κx) .
−∞
When we now substitute y = κx and use the familiar relation Hn (−x) = (−1)n Hn (x) of
the Hermite polynomials to rewrite the negative part of the integral, we arrive at
Ien,j,µ =
134
1
κµ−1
1 + (−1)µ+j
In,j,µ .
2
H.2. General expression
Therefore, Ien,j,µ vanishes if j is even and µ is odd or vice versa.
The remaining integral
In,j,µ
Z∞
2
≡ 2 dy y µ Hn+j (y) Hn (y) e −y
(H.4)
0
can be evaluated with the help of the Taylor series
Hn (x) ≡
n
X
an,α xα
α=0
of the Hermite polynomials. The expansion coefficients
(α)
an,α ≡
Hn (0)
α!
(H.5)
contain the αth derivative of Hn . Integration over y in eq. (H.4) yields
In,j,µ =
n
X
α=0
an,α
n+j
X
an+j,β Γ
β=0
µ+α+β+1
2
(H.6)
which vanishes for µ = 0 and j 6= 0 since the Hermite polynomials are orthogonal to
each other [50], that is
√
In,j,0 = π 2n−1 n! δj,0 .
Finally, we can express the matrix elements, eq. (H.3), by
∞
X
fµ 1 + (−1)µ+j
p
hn + j|f (x̂)|ni ≡ κ f0 δj,0 +
In,j,µ
2
2n 2j π n! (n + j)! µ=1 κµ
 ∞
P f2µ


In,j,2µ
for j even

κ2µ

κ2
µ=0
= p
(H.7)
∞
P
f2µ+1
2n 2j π n! (n + j)! 


I
for
j
odd

κ2µ+1 n,j,2µ+1
2
κ2
µ=1
in terms of In,j,µ , eq. (H.6). Since the Hermite polynomials Hn are either even or odd,
half of the coefficients an,α , eq. (H.5), are zero. Nevertheless, eq. (H.7) is quite useful to
calculate the matrix elements numerically.
135
H. Matrix elements
H.3. Explicit expression up to fourth order
This section contains the explicit expressions of the matrix elements calculated with a
polynomial operator fˆ of fourth order, used in section 8.2.2 to approximate the Riemann
Hamiltonian. For µmax = 4, eq. (8.16) and eq. (8.17) read
Fn(4) ≡ ε0 f˜0 +
and
Cn(4) ≡
respectively. The coefficients
4
X
j=1
4
X
j=1
h
i
ε(j+) Θ(n − j) f˜−j + f˜j
h
i
ε(j−) Θ(n − j) f˜−j − f˜j ,
2
f˜±j ≡ hn ± j|fˆ|ni
denote the matrix elements. With the help of tab. H.1, we get for the even coefficients
f˜0 =
f˜−2 =
f˜2 =
f˜−4 =
f˜4 =
2
3κ4
1
1
2
f2 +
n +n+
f4
f0 + κ n +
2
2
2
2
κ4
n(n − 1) f2 + κ2 (2n − 1) f4
4
2
κ4
(n + 1)(n + 2) f2 + κ2 (2n + 3) f4
4
κ8
n(n − 1)(n − 2)(n − 3) f42
16
κ8
(n + 1)(n + 2)(n + 3)(n + 4) f42 ,
16
2
that is polynomials of order O(n4 ), and for the odd coefficients
2
3κ2
κ2
˜
n f1 +
n f3
f−1 =
2
2
2
κ2
3κ2
˜
f1 =
(n + 1) f1 +
(n + 1) f3
2
2
κ6
n(n − 1)(n − 2) f32
f˜−3 =
8
κ6
f˜3 =
(n + 1)(n + 2)(n + 3) f32
8
polynomials of order O(n3 ).
136
Bibliography
[1] M. Planat, P. Solé, and S. Omar
Riemann hypothesis and quantum mechanics
J. Phys. A: Math. Theor. 44, 145203 (2011)
[2] E. Landau
Handbuch der Lehre von der Verteilung der Primzahlen (Chelsea Publishing
Company, New York, 1974)
[3] B. Riemann
Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse
Monatsberichte der Berliner Akademie (Nov. 1859)
Transcribed German version and English translation by D. Wilkins see
http://www.claymath.org/millennium/Riemann Hypothesis/1859 manuscript/
[4] E. C. Titchmarsh
The theory of the Riemann zeta-function (Clarendon Press, Oxford, 1967)
[5] A. A. Karatsuba and S. M. Voronin
The Riemann zeta-function (De Gruyter, Berlin, 1992)
[6] D. Schumayer and D. A. W. Hutchinson
Colloquium: Physics of the Riemann hypothesis
Rev. Mod. Phys. 83, 307 (2011)
[7] M. V. Berry and J. P. Keating
The Riemann zeros and eigenvalue asymptotics
SIAM Review 41, 236 (1999)
[8] G. Sierra and J. Rodrı́guez-Laguna
H = xp model revisited and the Riemann zeros
Phys. Rev. Lett. 106, 200201 (2011)
[9] R. K. Bhaduri, A. Khare, and J. Law
Phase of the Riemann ζ function and the inverted harmonic oscillator
Phys. Rev. E 52, 486 (1995)
[10] J.-B. Bost and A. Connes
Hecke algebras, type III factors and phase transitions with spontaneous symmetry
breaking in number theory
Selecta Mathematica 1, 411 (1995)
137
Bibliography
[11] F. Gleisberg, R. Mack, K. Vogel, and W. P. Schleich
Factorization with a logarithmic energy spectrum
New J. Phys. 15, 023037 (2013)
[12] C. Weiss, S. Page, and M. Holthaus
Factorising numbers with a Bose-Einstein condensate
Physica A 341, 586 (2004)
[13] R. Mack, J. P. Dahl, H. Moya-Cessa, W. T. Strunz, R. Walser, and W. P. Schleich
Riemann ζ function from wave-packet dynamics
Phys. Rev. A 82, 032119 (2010)
[14] J. Twamley and G. J. Milburn
The quantum Mellin transform
New J. Phys. 8, 328 (2006)
[15] C. Feiler and W. P. Schleich
Entanglement and analytical continuation: an intimate relation told by the
Riemann zeta function
New J. Phys. 15, 063009 (2013)
[16] E. T. Jaynes and F. W. Cummings
Comparison of quantum and semiclassical radiation theories with applications to
the beam maser
Proc IEEE 51, 89 (1963)
[17] H. Paul
Induzierte Emission bei starker Einstrahlung
Ann. Phys. 466, 411 (1963)
[18] E. P. Wigner
On the quantum correction for thermodynamic equilibrium
Phys. Rev. 40, 749 (1932)
[19] S. Wallentowitz, R. L. de Matos Filho, and W. Vogel
Determination of entengled quantum states of trapped atom
Phys. Rev. A 56, 1205 (1997)
[20] J. E. Moyal
Quantum mechanics as a statistical theory
Math. Proc. Camb. Phil. Soc. 45, 99 (1949)
[21] H. J. Groenewold
On the principles of elementary quantum mechanics
Physica 12, 405 (1946)
[22] C. Feiler
Zahlentheorie und Quantenmechanik - Die ζ-Funktion im Phasenraum
Diploma Thesis (2007)
138
Bibliography
[23] E. Schrödinger
Die gegenwärtige Situation in der Quantenmechanik
Die Naturwissenschaften 48, 807, ibid. 49, 823, ibid. 50, 844 (1935)
English translation in: J. A. Wheeler and W. H. Zurek
Quantum theory and measurement (Princeton University Press, Princeton, 1983)
[24] W. Schleich, M. Pernigo, and Fam Le Kien
Nonclassical state from two pseudoclassical states
Phys. Rev. A 44, 2172 (1991)
[25] M. Brune, E. Hagley, J. Dreyer, X. Maı̂tre, A. Maali, C. Wunderlich, J. M.
Raimond, and S. Haroche
Observing the progressive decoherence of the ‘meter’ in a quantum measurement
Phys. Rev. Lett. 77, 4887 (1996)
[26] S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and
S. Haroche
Reconstruction of non-classical cavity field states with snapshots of their
decoherence
Nature 455, 510 (2008)
[27] C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland
A “Schrödinger cat” superposition state of an atom
Science 272, 1131 (1996)
[28] W. P. Schleich
Quantum optics in phase space (Wiley-VCH, Berlin, 2001)
[29] C. L. Siegel
Über Riemanns Nachlaß zur analytischen Zahlentheorie
Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik,
Abt. B: Studien 2, 45 (1932)
[30] Clay Mathematics Institute
Millennium prize problems: Riemann hypothesis
http://www.claymath.org/millennium/Riemann Hypothesis/
[31] M. V. Berry and J. P. Keating
A new asymptotic representation for ζ( 12 + it) and quantum spectral determinants
Proc. R. Soc. Lond. A 437, 151 (1992)
[32] G. H. Hardy and J. E. Littlewood
The zeros of Riemann’s zeta-function on the critical line
Mathematische Zeitschrift 10, 283 (1921)
[33] E. Jahnke and F. Emde
Funktionentafeln mit Formeln und Kurven (Teubner, Leipzig, 1933)
139
Bibliography
[34] A. A. Utzinger
Die reellen Züge der Riemann’schen Zetafunktion (Dissertation, Zürich, 1934)
[35] A. Speiser
Geometrisches zur Riemannschen Zetafunktion
Math. Ann. 110, 514 (1935)
[36] J. Arias-de-Reyna
X-ray of Riemann zeta function
arXiv:math/0309433 (2003)
[37] K. A. Broughan and A. R. Barnett
The holomorphic flow of the Riemann zeta function
Math. Comp. 73, 987 (2003)
[38] J. W. Neuberger
Steepest descent and differential equations
J. Math. Soc. Japan 37, 187 (1985)
[39] J. W. Neuberger
Sobolev gradients and differential equations (Springer, Heidelberg, 1997)
[40] J. W. Neuberger, C. Feiler, and W. P. Schleich
Continuous Newton method, lines of constant phase and the Riemann zeta
function
To be published.
[41] J. W. Neuberger
Continuous Newton’s method for polynomials
Math. Intell. 21, 18 (1999)
[42] N. Levinson and H. L. Montgomery
Zeros of the derivatives of the Riemann zeta-function
Acta mathematica 133, 49 (1974)
[43] S. Haroche and J.-M. Raimond
Exploring the quantum: atoms, cavities and photons (Oxford University Press,
Oxford, 2006)
[44] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland
Quantum dynamics of single trapped ions
Rev. Mod. Phys. 75, 281 (2003)
[45] M. O. Scully and M. S. Zubairy
Quantum optics (Cambridge University Press, Cambridge, 1997)
[46] J. P. Dowling
A quantum state of ultra-low phase noise
Opt. Comm. 86, 119 (1991)
140
Bibliography
[47] Wolfram Research, Inc.
Mathematica Version 8.0 (Champaign, Illinois, 2010)
[48] K. Vogel, V. M. Akulin, and W. P. Schleich
Quantum state engeneering of the radiation field
Phys. Rev. Lett. 71, 1816 (1993)
[49] A. S. Parkins, P. Marte, P. Zoller, and H. J. Kimble
Synthesis of arbitrary quantum states via adiabatic transfer of Zeeman coherence
Phys. Rev. Lett. 71, 3095 (1993)
[50] I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig
Taschenbuch der Mathematik (Harri Deutsch, Frankfurt am Main, 2000), 5 edn.
[51] Z.-H. Li, D.-W. Wang, H. Zheng, S.-Y. Zhu, and M. S. Zubairy
Effect of the counterrotating-wave terms on the spontaneous emission from a
multilevel atom
Phys. Rev. A 80, 023801 (2009)
[52] R. L. de Matos Filho and W. Vogel
Engineering the Hamiltonian of a trapped atom
Phys. Rev. A 58, R1661 (1998)
[53] W. Vogel and R. L. de Matos Filho
Nonlinear Jaynes-Cummings dynamics of a trapped ion
Phys. Rev. A 52, 4214 (1995)
[54] P. H. Fuss (editor)
Correspondance mathématique et physique de quelques célèbres géomètres du
XVIIIème siècle (Saint Petersburg, 1843)
Volume 1: Letter from L. Euler to C. Goldbach (1729)
Volume 2: Letter from D. Bernoulli to C. Goldbach (1729)
[55] E. T. Whittaker and G. N. Watson
A course of modern analysis (University Press, Cambridge, 1969)
[56] M. Abramowitz and I. A. Stegun
Handbook of mathematical functions (Dover Publications, New York, 1972), 9 edn.
[57] A. Sommerfeld
Elektrodynamik (Verlag Harri Deutsch, Thun-Frankfurt, 1988)
[58] A. M. Odlyzko
The 1022 -nd zero of the Riemann zeta function
Contemp. Math. 290, 139 (2001)
Tables of zeros of the Riemann zeta function see
http://www.dtc.umn.edu/∼odlyzko/zeta tables/
141
Bibliography
[59] X. Gourdon and P. Demichel
The 1013 first zeros of the Riemann Zeta function, and zeros computation at very
large height (2004)
http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e131e24.pdf
[60] K. Husimi
Some formal properties of the density matrix
Proc. Phys.-Math. Soc. Japan 22, 264 (1940)
[61] Y. Kano
A new phase-space distribution function in statistical theory of the electromagnetic
field
J. Math. Phys. 6, 1913 (1965)
[62] R. J. Glauber
Photon correlations
Phys. Rev. Lett. 10, 84 (1963)
[63] E. C. G. Sudarshan
Equivalence of semiclassical and quantum mechanical descriptions of statistical
light beams
Phys. Rev. Lett. 10, 277 (1963)
[64] D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J.
Wineland
Experimental determination of the motional quantum state of a trapped atom
Phys. Rev. Lett. 77, 4281 (1996)
[65] C. Kurtsiefer, T. Pfau, and J. Mlynek
Measurement of the Wigner function of an ensemble of helium atoms
Nature 386, 150 (1997)
[66] M. Freyberger, P. J. Bardroff, C. Leichtle, G. Schrade, and W. P. Schleich
The art of measuring quantum states
Phys. World 10, 41 (Nov. 1997)
[67] M. B. Plenio and S. Virmani
An introduction to entanglement measures
Quant. Inf. Comput. 7, 1 (2007)
[68] M. A. Nielsen and I. L. Chuang
Quantum computation and quantum information (Cambridge University Press,
Cambridge, 2000)
142
Zusammenfassung
Die Riemannsche Zetafunktion ζ bildet durch ihre enge Verknüpfung mit der Zahlentheorie nicht nur eine wesentliche Grundlage der Mathematik, sondern beeinflusst auch die
Physik. Deshalb ist es nicht verwunderlich, dass neben mathematischen Ansätzen zum
Beweis der Riemannschen Hypothese, welche die Lage der sogenannten nicht-trivialen
Nullstellen von ζ beschreibt, auch physikalische Methoden herangezogen werden [6].
Ziel dieser Arbeit ist es, durch verschiedene mathematische wie physikalische Darstellungen der Zetafunktion Einblicke in ihr Verhalten zu erlangen. Dazu werden im ersten
Kapitel die benötigten mathematischen Formeln zusammengetragen, die ζ exakt oder
näherungsweise beschreiben.
Das zweite Kapitel widmet sich der Darstellung der Zetafunktion mit Hilfe der
kontinuierlichen Newton-Methode. Die daraus erhaltenen Newtonschen Flüsse veranschaulichen die Topologie der Zetafunktion durch Kurven konstanter Phase, welche die
Lage des einzigen Pols sowie der Nullstellen der Funktion und ihrer ersten Ableitung in
der komplexen Ebene besonders hervorheben. Deshalb wäre auch eine Verletzung der
Riemann-Hypothese im Newtonschen Fluss sichtbar.
In den folgenden Kapiteln wird ein physikalischer Zugang, basierend auf dem Überlapp
hΦ|Ψ(t)i zweier quantenmechanischer Zustände, vorgestellt. Dabei werden die Phasen
in den Summanden der verschiedenen Darstellungen von ζ mit Hilfe der Zeitentwicklung des verschränkten Zustands |Ψ(t)i realisiert, welcher analog den Zuständen des
Jaynes-Cummings-Paul-Modells gewählt wurde: Das Feld eines Hohlraumresonators,
beschrieben durch eine Superposition von Fock-Zuständen, wird mit einem Zwei-NiveauAtom gekoppelt. Die Wechselwirkung wird allerdings durch den Riemann-Hamiltonian
ĤR ≡ ~ω ln(n̂ + 1) σ̂z gesteuert, der neben dem Pauli-Operator σ̂z , welcher das Vorzeichen der auftretenden Phasen bestimmt, eine logarithmische Abhängigkeit vom Photonenzahloperator n̂ aufweist.
Es stellt sich im vierten Kapitel heraus, dass für Argumente s ≡ σ + iτ von ζ mit
Realteil σ größer als eins auf Verschränkung mit dem Zwei-Niveau-Atom verzichtet werden kann. Die Riemann-Zustände |σ, τ i, die lediglich das Hohlraumfeld beschreiben,
genügen, um die Dirichlet-Summe zu erzeugen, da diese nur die Phasen −iτ ln(n + 1)
aufweist. Allerdings sind die Riemann-Zustände aufgrund ihrer Normierung ebenfalls
auf den Bereich σ > 1 beschränkt.
Mathematisch lässt sich der Definitionsbereich mittels einer alternierenden Summe auf
alle positiven σ erweitern. Die daraus resultierenden Zustände sind aber weiterhin auf
den Bereich σ > 1 beschränkt, solange man die exakte, also unendliche Summe darstellen
möchte. Selbst Verschränkung hilft in diesem Fall nicht. Erst die Beschränkung auf eine
endliche Anzahl an Fock-Zuständen erzielt das gewünschte Ergebnis. Die abgeschnittenen Riemann-Zuständen |σ, τ iν sind auch im Bereich 0 < σ < 1 definiert. Jedoch ist die
143
Zusammenfassung
Näherung der Zetafunktion durch eine abgeschnittene alternierende Summe für kleine
Abschneideparameter nicht besonders gut. Große Abschneideparameter führen hingegen zur Verkleinerung der Wahrscheinlichkeitsamplituden der einzelnen Fock-Anteile und
somit zu sehr kleinen Überlappwerten.
In den Kapiteln 6 und 7 verwenden wir deshalb die Riemann-Siegel sowie die BerryKeating-Formel als Näherung für ζ. Beide Darstellungen bestehen aus zwei endlichen
Summen mit relativ kleiner Summationsgrenze, in deren Summanden sowohl die Phasen
−iτ ln(n+1) als auch deren komplex Konjugierte +iτ ln(n+1) auftreten. Dies macht die
Verwendung von verschränkten Zuständen zur physikalischen Darstellung von ζ nötig.
Die Beträge der einzelnen Summanden unterscheiden sich in den beiden Näherungen
nur minimal. Trotzdem führt dieser kleine Unterschied zu einer erheblichen Verbesserung
der Näherung durch die Berry-Keating-Formel, die bereits den ersten Korrekturterm zur
Riemann-Siegel-Formel enthält. Des Weiteren unterscheiden sichhdie Summationsgreni
p
zen: In der Riemann-Siegel-Formel wird bis zu einem Wert n0 ≡
τ /(2π) summiert,
wobei mit [. . . ] die Gaußklammer gemeint ist. Deshalb springen die Funktionswerte
sobald τ = 2π n20 erreicht ist. Diese Diskontinuitäten werden durch die Berry-KeatingFormel vermieden, deren Summationsgrenze lediglich nB ≥ n0 erfüllen muss.
Bei der physikalischen Darstellung der beiden Formeln machen wir uns ihre Ähnlichkeit zu Nutze. Zur Realisierung der Phasenfaktoren genügt die Zeitentwicklung des
Riemann-Siegel-Zustands |ΨRS(t)i, der an die Darstellung von Schrödingerkatzen erinnert. Wir koppeln allerdings zwei entgegengesetzt rotierende abgeschnittene RiemannZustände mit den beiden Atomzuständen, da diese eine logarithmische anstelle einer linearen Abhängigkeit der Phase von der Photonenzahl n aufweisen. Durch die Zeitentwicklung wird der anfangs unverschränkte Zustand |ΨRS(0)i verschränkt. Im Phasenraum
wirkt sich die Zeitentwicklung auf unterschiedliche Weise aus: Die Diagonalelemente der
Wignermatrix Ŵ|ΨRSi rotieren ebenfalls in entgegengesetzte Richtungen, wohingegen die
Nichtdiagonalelemente die anfängliche Symmetrie in p beibehalten und nur ihre interne
Struktur verändern.
Die Unterschiede der beiden Näherungen werden im physikalischen Bild durch Verwendung unterschiedlicher Referenzzustände erreicht. Der Riemann-Siegel-Referenzzustand
|ΦRSi besteht aus den abgeschnittenen Riemann-Zuständen |σ, 0in0 und |2 − 3σ, 0in0 gekoppelt an das Atom. Dadurch ist der Zustand für σ 6= 1/2 verschränkt und wird nur
auf der kritischen Achse, also für σ = 1/2, unverschränkt. Im Phasenraum sind deshalb
zwar die Diagonalelemente der Wignermatrix Ŵ|ΦRSi für alle σ symmetrisch in p, die
Nichtdiagonalelemente hingegen nur für σ = 1/2.
Im Gegensatz dazu setzt sich der Berry-Keating-Referenzzustand |ΦBKi aus den beiden
Berry-Zuständen |φB(σ, τ )inB und |φB(2−3σ, τ )inB zusammen, die sich nur sehr wenig von
den abgeschnittenen Riemann-Zuständen zur Zeit τ = 0 unterscheiden. Dennoch führt
der kleine Unterschied dazu, dass |ΦBKi auch auf der kritischen Achse verschränkt ist und
lediglich die Nichtdiagonalelemente der Wignermatrix Ŵ|ΦBKi für σ = 1/2 symmetrisch
in p sind.
Nichtsdestotrotz zeigt die kritische Achse in den Phasenraumbildern beider Näherungen ihre besondere Rolle: nur für σ = 1/2 ist der Integralkern der Wignerdarstellung K
144
sowie der Betrag des Integralkerns in Moyaldarstellung K symmetrisch in p.
Im letzten Kapitel wird gezeigt, dass die Ähnlichkeit zwischen dem RiemannHamiltonian und dem effektiven Hamiltonian des Jaynes-Cummings-Paul-Modells
zur Berechnung eines genäherten Hamiltonian verwendet werden kann. Eine grobe
Abschätzung mit Hilfe eines Polynoms vierter Ordnung führt zu einer guten Simulation
des allgemeinen Verhaltens der Riemannschen Zetafunktion obwohl der genäherte
Hamiltonian vom Riemann-Hamiltonian abweicht.
Abschließend bleibt zu erwähnen, dass unsere physikalische Beschreibung der Riemannschen Zetafunktion auch auf komplexe Funktionen f (s) mit Argument s ≡ σ + iτ
angewendet werden kann, wenn sich diese als Reihe
f (σ + iτ ) ≡
∞ X
an (σ, τ )
bn (σ, τ )
+
(n + 1) iτ
(n + 1)−iτ
n=0
darstellen lassen. Wie bei ζ können auch hier die Faktoren (n + 1)±iτ als Zeitentwicklung eines verschränkten Quantenzustands |Ψ(t)i interpretiert werden, wobei die Wechselwirkung wieder durch ĤR beschrieben wird, also logarithmisch vom Photonenzahloperator n̂ abhängt. τ = ωt ist hier die skalierte Zeit. Die Faktoren an und bn werden auf die Wahrscheinlichkeitsamplituden des Zustands |Ψ(t)i und des entsprechend
gewählten Referenzzustands |Φi aufgeteilt. Hierbei ist darauf zu achten, dass die Wahrscheinlichkeitsamplituden des Anfangszustands |Ψ(0)i zeitunabhängig sind, da sonst die
Phasen nicht durch die Zeitentwicklung mit ĤR entstehen. Folglich müssen alle τ -Abhängigkeiten der Faktoren an und bn in den Referenzzustand einfließen, da dort τ = Im s
lediglich den Imaginärteil des Punktes s im Phasenraum darstellt.
Auf diese Weise können die Eigenschaften der Funktion auf die Zustände transferiert
und für Messungen zugänglich gemacht werden.
145
Publikationen
1. E. Kajari, M. Buser, C. Feiler und W. P. Schleich
Rotation in Relativity and the Propagation of Light
in Atom Optics and Space Physics, Proceedings of the International School of
Physics “Enrico Fermi”, Course CLXVIII, edited by E. Arimondo, W. Ertmer,
W. P. Schleich and E. M. Rasel (Elsevier, Amsterdam, 2009)
ebenfalls erschienen in:
Rivista del nuovo cimento 32, 339 – 437 (2009)
2. C. Feiler, M. Buser, E. Kajari, W. P. Schleich, E. M. Rasel and
R. F. O’Connell
New Frontiers at the Interface of General Relativity and Quantum Optics
Space Sci Rev 148, 123 – 147 (2009)
ebenfalls erschienen in:
Probing the Nature of Gravity – Confronting Theory and Experiment in Space,
Editors: C. W. F. Everitt, M. C. E. Hulrt, R. Kallenbach, G. Schäfer, B. F. Schultz
und R. A. Treumann (Springer, New York, 2010)
3. S. Wölk, C. Feiler and W. P. Schleich
Factorization of numbers with truncated Gauss sums at rational arguments
J. Mod. Opt. 56, 2118 – 2124 (2009)
4. S. Wölk, C. Feiler and W. P. Schleich
Quantum Mechanics Meets Number Theory
Conference Paper zu International Conference on Quantum Information, Ottawa,
Kanada (Optical Society of America, 2011)
5. C. Feiler and W. P. Schleich
Entanglement and analytical continuation: an intimate relation told by the Riemann zeta function
New J. Phys. 15, 063009 (2013)
6. J. W. Neuberger, C. Feiler, H. Maier and W. P. Schleich
Continuous Newton Method, Lines of Constant Phase and the Riemann Zeta Function
To be published.
147
Poster und Vorträge
1. C. Feiler, R. Mack and W. P. Schleich
Wignerfunktion zur Beschreibung der Riemann-Zeta-Funktion“
”
Poster bei der DPG-Tagung in Düsseldorf (März 2007)
2. C. Feiler, R. Mack and W. P. Schleich
Riemanns Zeta Function in Phase-Space“
”
Poster bei der DPG-Tagung in Darmstadt (März 2008)
3. C. Feiler, R. Mack and W. P. Schleich
The Riemann ζ-Function in Phase Space“
”
Poster bei der DPG-Tagung in Hamburg (März 2009)
4. C. Feiler, R. Mack and W. P. Schleich
Zeta-States in Phase Space“
”
Poster bei der DPG-Tagung in Hannover (März 2010)
5. C. Feiler
Riemann zeta function in phase space“
”
Vortrag bei der Netzwerktagung der Alexander von Humboldt-Stiftung in Ulm
(Oktober 2010)
6. C. Feiler, R. Mack and W. P. Schleich
Zeta-States in Phase Space“
”
Poster bei der Conferece on Quantum Simulations“ in Benasque (Spanien,
”
Februar/März 2011) und der DPG-Tagung in Dresden (März 2011)
7. C. Feiler
Newton flow of the Riemann zeta function“
”
Vortrag beim Ulm-Augsburg Meeting 2013 in Augsburg (April 2013)
149
Curiculum vitae
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151
Danksagung
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153
Erklärung
Hiermit versichere ich, dass ich die vorliegende Dissertation selbständig verfasst und
keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe, sowie die
wörtlich oder inhaltlich übernommenen Stellen als solche kenntlich gemacht habe.
Ulm, den
Cornelia Feiler