On Prime Numbers and Riemann Hypothesis
by L. M. Ionescu
Department of Mathematics
Illinois StateUniversity
Abstract. A partial ordered set structure on the prime numbers was recently discovered
by the PI. This “hierarchy” opens a new perspective on the suspected duality between prime
numbers and non-trivial zeroes of the Riemann zeta function, as an algebraic approach for
the celebrated Riemann Hypothesis.
In order to address these fundamental problems, the proposed research targets: 1) understanding the role of this POSet structure on the duality theory of algebraic group associated
to the rational numbers, viewed as a discrete group; 2) understanding the relation between
the above duality and the “classical” p-adic duality of rationals, via the field of adeles, when
“removing” the continuum of real numbers. 3) The study of the role of the absolute Galois
group, in the above duality;
4) Apply, or at least compare, the above duality with the Riemann-Siegel formula and
Gurzweil trace formula, as shadows of dualities associated with Dirichlet series, L-series and
discrete Melin transforms.
The above problematics has a tempting quantum physics interpretation, in terms of
Riemann gas / Primon model, with Riemann Zeta function as a canonical partition function,
and its non-trivial zeros as scatering amplitudes. The PI intends to investigate further
these interpretations by using techniques from quantum groups and deformation theory;
specifically, vieweing the primes as generators of rationals, and applying Milnor-Moore
duality as in the work of Coones-Kreimer on algebraic renormalization, with the POSet as
an analog of Kontsevich / Feynman graphs.
Finally, the connections between Multiplicative Number Theory and Quantum Physics
will be developed, starting with the correspondence between fine structure constant and
Riemann zeta function value for s = 2 , and the approximate correspondence between
quark masses and Fermat primes as the building blocks of the prime numbers under the
POSet structure.
1
2
Contents
1. Introduction
1.1. Klein Geometry of Finite Fields and The POSet of Primes
1.2. Lie Theory Approach and Algebraic Quantum Groups
1.3. The Prime Numbers - Rooted Trees Correspondence
1.4. Primes as Rooted Trees and The Universal Cohomological Problem
1.5. A Homological Interpretation of Primes and Riemann-Roch Duality
1.6. Prime Trees and Primality Tests
1.7. The Distribution of Prime Numbers: Integers vs. Rooted Trees
1.8. Additive versus Multiplicative Number Theory; Graded Structures
2. On Duality of Rationals, Adels and Dirichlet Series
2.1. On Duality: additive vs. multiplicative
2.2. Adels
2.3. Dirichlet Series
2.4. The Riemann Zeta Function and the POSet of Primes
3. Dirichlet Series, Melin Transform and The Elliptic Curves Paradigm
3.1. Riemann-Mangoldt Formula and Parceval Theorem
3.2. Primon Model and Breaking the Symmetry
4. Riemann-Siegel formula and Gutzwiller trace formula
5. Plan of Research and Development
6. Conclusions and Further Developments
6.1. Multiplicative Number Theory as Ultimate Physics Theory
6.2. Understanding Multiplicative Number Theory First!
References
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1. Introduction
Kroneker supposedly said “God made the natural numbers; all the rest is work of man”.
Pondering on addition, and Additive Number Theory vs. Multiplicative Number Theory,
we realize that counting is human, and that God had to make the prime numbers only ...
but how?
The mystery of prime numbers is an ancient one; and was occasionally considered a
“problem”, until the emphasis was put on their distribution, leading to Riemann’s approach
to the Prime Number Theorem via his celebrated zeta function:
∞
X
ζ(s) =
n−s .
1
It igniting an intense research aiming to prove that the non-trivial zeroes of its analytic
continuation lie on the critical line s = 1/2 + it [1].
Both problems, the nature of prime numbers and Riemann Hypothesis are, no doubt,
the most important mathematical problems, historically and from a practical point of view
(e.g. RSA encryption schemes).
The PI’s interest in the prime numbers and Riemann zeta function arose from the “physics
side” [2]; the PI’s initial investigation from a geometric point of view reveled a partial order
on the set of prime numbers (POSet). This is not only relevant to their distribution within
the natural numbers, but also to the presumed duality between the prime numbers and the
non-trivial zeroes of the Riemann zeta function.
In what follows we will present PI’s current understanding of this framework, interconnections with Riemann hypothesis and the various threads of research which stem from
them.
1.1. Klein Geometry of Finite Fields and The POSet of Primes. To understand
the prime numbers, which are a shadow of basic finite fields, we should understand their
algebraic structure, i.e. their symmetries.
From an algebraic point of view, the prime numbers are the generators of integers; but
what are their “generators” in the sense of Lie-Klein, i.e. their symmetries?
If p is a prime number and Fp the corresponding finite field, its symmetries are its
automorphisms:
prime number : p → f inite f ield Fp → symmetries : Aut(Fp ) ∼
= F∗p ∼
= ⊕Z/q k Z,
Y
|F∗p| = p − 1 = 2k
qiei .
Adopting the point of view of Felix Klein, we have a geometry (Fp , Aut(Fp)) , which breaks
down the space into minimal orbits, the equivalent of prime ideals for an action.
These orbits, or building blocks of primes, correspond to p-groups Zkq , which are p-adic
integers truncated at depth k .
Forgetting the depth for now, it defines a partial order p > q , for those primes q dividing
p − 1 , i.e. the generators of symmetries of p .
These leads to a descending (almost) hierarchy of primes, leading from more “complex
primes”, to “simpler primes”.
For example, p = 13 “contains within itself” the simpler primes 2 and 3 , as generators
of its symmetries: 13 − 1 = 22 · 3 . So, the numerical factorization 13 − 1 = 22 · 3 , which
we will call it the Proth factorization, reflects the structure of the symmetries of the finite
field F13 .
4
Conceptually, it structures the set of primes as a partial ordered set (not an infinite tree
though), with the known Fermat primes (conjectured to be all)
n
Fk = 22 + 1,
n = 0, 1, 2, 3, 4
at its “boundary”, as irreducible building blocks. These “indivisible primes” are the analog
of quarks for elementary particles, noted by the PI as the 2nd amazing coincidence [2],
and supporting the 1st, regarding the relation between the zeta function and fine structure
constant.
This is, as far as the PI could find in the current literature, the first investigation in
the “nuclear mathematics” of finite fields, as opposed the usual direction of building bigger
finite fields starting from the basic finite fields (cyclotomic extensions as a “chemistry” of
finite fields).
The results of Proth [3, 4, 5], regarding similar factorizations, although relevant, are at
the present stage only numerical shadows of possibly important clues about the internal
structure of prime numbers.
For example, with the atomic primes pictured as a boundary (only three of the 5 Fermat
primes are shown below, together with the “degenerate” Fermat prime 2), the lower portion
of the POSet of primes P is:
1979E
Level 3 :
Level 2 :
Level 1 :
F ermat P rimes
29/ @
//@@@
// @@
// @@
// 7
//
//
/ 2
EE
EE
EE
E"
23 OOO
OOO
{{
zz
{
z
OOO
z
{{
z
OOO
{
z
{
z
OO'
z
}{{
z
zz
31
11
CC
z
@
y
@@
CC zzz
~~
yy
@@
~
CzCz
y
y
@@
~~
zz CCC yyy
@ ~~~
! |y
|zz
43
3
5
17 · · ·
The power k of 2, representing the binary content ( p = 2k n + 1 ), is encoded as labels of
arrows, and can be interpreted (for now) as the difference of “energy” levels; for example
29 = 1 + 22 · 7 sits at level of 7 plus two “stories” higher, while 31 = 1 + 2 · 3 · 5 is one
level higher then 3 and 5 . The arrows between Fermat primes were omitted for clarity.
The POSet contains unoriented cycles (not simply connected, i.e. not tree-like), like the
above triangle with vertices 43, 7 and 3 , and it is capable of forming “proper modes” /
periods:
p → Hp → Spec(Hp), En = ~ωn , length ∼ f requency,
where
X
X
length =
orbits =
ek pk
is a Lie “algebra” cochain (additive number theory; multiplicative partitions of n ), e.g. Zpe
has length log pk = k log p .
A natural question arises at this stage:
Question 1. Do these periods of the convolution algebra of the POSet of prime numbers
(reminescent of Hodge periods), have anything to do with the Argand cycles of the Rieman
zeta function restricted to the critical line.
The connection is also prompted by the know interpretation of zeta zeroes as scatering
amplitudes [15].
5
1.2. Lie Theory Approach and Algebraic Quantum Groups. The prime 2 is extraspecial: not only the only even, and sort of a degenerate fermat prime F−∞ , but it in fact
is a unit in the “tangent space” at 1 (Lie module of the discrete group of positive rationals
( Q×
+ , ·, 1) :
Exp(Xp starXq ) = (p − 1)(q − 1),
Exp : gP → Q+ .
Here P is the POSet of primes, gP is the free Lie BZ -module (abelian group) and the
exponential is the tangent map Exp(Xp) = p − 1 . We will refer to these context as the Lie
Theory interpretation of the rationals.
Indeed X2 ? Xq = Xq ; we think of q − 1 as a position vector, and ? is the usual
star-product associated to multiplication:
Exp(Xp ? Xq ) = Exp(Xp) · Exp(Xq ).
Now the Lie module (algebra) of primes has a contraction:
X
Y e
h : g → g, h(Xp ) =
ei Xqi , p − 1 =
qi i , Ex. : h(X7 ) = X2 + X3 , h(X5 ) = 2X2 ,
relative to the grading given by the “depth of the prime tree” §1.3. It allows to “extend” the
Lie theoretic approach, and apply the Theory of Deformations [26] (more later on, e.g. §1.5),
towards a Quantum Geometry of multiplicative integers, useful for physics applications §6.1.
The geometric-algebraic significance of these facts will be subject to future investigations;
in some sense 2 plays the role of 0 , as a prime ideal of the spectrum in “additive” algebraic
geometry.
The above Lie Theory interpretation allows to apply adapt the results of Milner-Moore
Theorem regarding the Lie algebra - Universal Enveloping Algebra duality to the case of
the Lie module of prime numbers, the rational numbers as a discrete group and the duality
theory of finitely suported arithmetic functions kf in Q+ , viewed as an algebraic quantum
group [19]. Briefly said, an algebraic quantum group is the extension of the Hopf algebra
concept, when allowing multipliers as elements of the dual, instead of the usual reduced
dual which does not give a good duality theory.
In this way, we enable the powerfull machinery of duality, with prospects of “correcting”
the classical duality of the rationals, which involves the reals:
Q+ → QA → Q̂.
Here QA is the group of adeles and Q is the group of characters of the rationsls, viewed
as a discrete group.
More will be said later on, including its applications to Riemann hypotheis and absolute
Galois group.
1.3. The Prime Numbers - Rooted Trees Correspondence. Each prime number
defines an associated a canonical rooted tree, corresponding to its POSet subgraph [2] (see
also [8]):
Ψ : P → RT ,
trees representing the possible paths to the “final prime” 2 .
6
For example the trees first few primes are:
2
3
◦
◦
◦
◦
~
~~
~
~
~
~
~
~~
~
~
~
~
•
5
•
7
•
•
•
•
11
◦
||
||
|
|
~|
|
•
•
|
||
||
|
~||
•
13
17
◦
◦
||
||
|
|
~|
|
21
•
22
•
•
•
•
For p = 13 the label indicates the multiplicity of the downarrow, i.e. two such arrows
descend from the parent node. For the other Fermat prime p = 17 , four such arrows
descend, i.e. fermat primes are corrolas.
The hierarchy trees yielding prime numbers are called prime trees. As a special case, the
prime trees representing Fermat primes, which are prime numbers of the form 2k+1, where
k=2n, will be called Fermat trees. These are trees with one node 2n.
The above defined prime trees correspond to a generalization of Proth primes [3], which
are primes of the form 2kn + 1 , with 2k > n . Some additional examples were computed
as part of a research project with students of the ISU Summer Research Academy [16].
Additional info on large primes of the form 2kn + 1 is provided in [4, 5]; the significance
of those results in the context of the POSet of primes, and especially of finite fields, will be
studied by the PI.
There is also a natural mapping from the rooted trees to the Lie module of primes,
Ψ∗ : RT → gP ,
which is a section for the embedding of the prime numbers as rooted trees. It has a natural
extension to the Hopf algebra of forests, which are disjoint unions of rooted trees.
This mapping associates to a rooted tree the formal sum of prime factors of the integer
obtained by reversing the Proth decomposition: at each level, multiply the values of the
“packed nodes” and add 1. For example the Φ∗ -value of the following tree 22 ·72 +1 = 3·131
is not a prime (multiplicity of a discendent is indicated as a label on the arrow):
◦

 2


 
•
•
•
•
•
t◦
tt
t
t
2
tt
tt
t
zt
2
u•
uu
u
u
uu
uu
u
z
u
2
2
=
2
◦
ppp
p
p
pp
2
ppp
p
p
x
p
2
=
2
=
1 + 22 · 72
7
1+2 = 3
= 393
The concept of fusion rules and multiplicity comes to mind: another subject to be investigated.
There are several Hopf algebra structures defined on rooted trees, including the ConnesKreimer Hopf algebra of rooted trees which is relevent to algebraic renormalization. Rooted
7
trees are also fundamental in the theory of free Lie algebras. This makes the above correspondence a very interesting object of study, allowing to pull-back many results regarding
rooted trees, as well as info regarding their distribution.
1.4. Primes as Rooted Trees and The Universal Cohomological Problem. At this
point we only mention as an important aspect to be studied the relationship between the
(graded) Hopf algebra generated by primes HP = C[P] and the Connes-Kreimer Hopf
algebra of rooted trees (Recall that in a graded bialgebra, the antipode comes “for free”).
The Proth decomposition of primes corresponds to the deletion of the root of prime trees,
i.e. rotted trees representing primes.
The adjoint at teh level of the rooted trees is the root adjoining ooperator B + on the
Hopf algebras of rooted trees.
It is interesting to study the relation with the universal cocycle L of Connes’ Universal
cohomological problem in non-commutative geometry [18], p.601:
B + (t1 , ..., tn) = root− > (t1 , ..., tn).
This could clarify how to construct more complex prime numbers.
For example, adjoining a root to the forest of rooted Q
trees of p = 3 and p = 5 yields
Q
another prime, if the construction/evaluation rule p−1 = qi is amended to p−1 = 2 qi ,
to take into account that p − 1 is always even:
B + (t3 , t5 ) = t31 ,
31 − 1 = 2(3 · 5).
This “amendment” is not ad-hoc, since finite fields, viewed as roots of unity, posses a natural
symmetry by conjugation, and they appear as “doubles” (projective connes). Moreover, the
role of the 2k factors as separate of the other odd factors of p − 1 , which is essential in the
theory of Proth primes, needs to be investigated in this context, from a conceptual point of
view.
1.5. A Homological Interpretation of Primes and Riemann-Roch Duality. At the
level of the Lie module (algebra) of primes g = Z{Xp}p∈P , the “adjoint” of B + is the the
contraction
X
Y
h(Xp) =
eq Xq , p =
qe,
corresponding to the Proth factorization of p , or algebraically, corresponding to the decomposition of the symmetry group of the basic field Fp into p-adic groups.
These abelian p-groups are trucations of p-adic integers, and should be thought of as n th
order deformations of the corresponding basic finite fields (tangent spaces/ Lie algebras).
For duality purposes, together with the finite fields we should consider also their duals
Fp ∗ , which may be thought to correspond to the negative primes (double of Q+ ).
Now the paradigm of deformation theory can be applied: the universal solution of the
maurer-Cartan equation [?, ?, ?].
The context here is that of a formal group, say QG , with a formal connection “a la”
Chen [?]. We expect at this stage that the finite fields, as p-cycles, to correspond somehow
to the homology basis of the fundamental space π1 (QG) .
The grading / hierarchy (torsors Tp = Aut(Fp ), Fp ) suggests that, maybe we are dealing
here with the full sequence of homotopy groups (non-abelian / generalized cohomologies
[?]).
The relation with the non-trivial zeroes of the Riemann zeta functions should fall within
an analog version of Riemann-Roch Theorem.
8
The relation with the familiar process of Eratostene’s sieve is given by the algebraicgeometry interpretation of primes as points of the spectrum, and the elimination of multiples
of the first few primes, as a “canonical divisor” (maybe), with furthere homology produced
on the open Zariski complement. The topological description of π1 of a universal cover,
as an “analitic continuation” of open simply connected neighborhoods comes to mind [?].
Then the natural candidate as a covering space is the Klein geometry of a finite field (torsor)
Tp .
The framework for implementing these ideas is the homotopy theory of the category of
finite sets (Sets, ⊕, ⊗, Aut()) , a shadow of addition, multiplication and the POSet structure
of primes. The finite fields apear as part of the center:
Aut(Fp ) ⊂ AutSet (Fp ) = Sp ,
where Sn are permutation groups.
The universal cover of the “pointed Riemann surface” which should explain the primes
as homology classes and zeta zeroes as periods, should have the deck transformations the
absolute Galois group, if we “match” their relevance to multiplicative number theory.
In conclusion, the existence of two such fundamental Hopf algebras, corresponding to
primes and rooted trees, must have a deeper meaning, towards the clarification of what
prime numbers really are. The universal cohomology problem for rooted trees invites to
consider a generalized cohomology interpretation of the primes based on the tautological
fibration Tp , towards a Riemann-Roch duality between primes and zeta function zeroes, and
a realization of the absolute Galois group as the group of the corresponding deck transformations.
1.6. Prime Trees and Primality Tests. The fact that prime numbers, or rather the corresponding basic finite fields, have a hierarchic internal structure opens a research direction
in the direction of primality tests.
For example Euler’s Theorem is used for various primality tests; how can the automorphisms of Fp be used in refining those tests?
More specifically, the Rabin-Miller Test for Composite Numbers [22], p.233, uses a property of prime numbers represented in “Proth form”: p − 1 = 2k q . It is natural to try
to push the analysis further down the hierarchy of the corresponding (prime) rooted tree,
when further decomposing q according to its symmetries.
Another clear direction of investigation consists in combining a test of primality with a
sieve based on the inverse mapping, from rooted trees to natural numbers.
1.7. The Distribution of Prime Numbers: Integers vs. Rooted Trees. Now, prime
numbers should not be considered in isolation: they are part of a partial ordered set.
Their distribution as elements of natural numbers appear to be random (Gaussian Unitary
Ensamble), but viewed as a POSet, with several natural grading functions, has a good
prospect of clarifying this fact.
This new structure on the set of prime numbers is like a 2D-resolution of the usual
linear 1D-order on the prime numbers derived from the order of natural numbers. The
importance of the additional information revealed is similar to the difference between a
TV-image flattened as a horizontal line, as opposed to the actual 2D-image.
There is an extensive knowledge about the distribution of rooted trees with various
labelings, and it is natural to expect that these results would shear additional light on
9
the old problem of distribution of prime numbers, as natural numbers (analitic aspects /
measure theory side).
A study of the distribution of prime numbers as rooted trees, within the graded set of
rooted trees, was already done as part ISU Summer Research Academy [17]. The grading
allows for a better “resolution” (2D) of how primes are distributed in a set with more
structure then the 1D, “collaped” set of additive integers.
1.8. Additive versus Multiplicative Number Theory; Graded Structures. It is
worth emphesizing that the relevant context for studying primes is NOT the additive integers, which reflect a measure theory point of view: a natural number, in view of its
factorization into primes, can be thought of as the total size (hyper-volume) of products
of sets, as an application of Fubini Theorem for cardinal measure; the prime numbers are
rather the simple objects within the multiplicative structure.
Their relevent distribution, therefor, is not within the integers, where they apear in a
random way, but rather within the POSet. Their “relative abundance” is controlled by the
“ L∞ -fusion rule”:
Y
q1e1 ...qlel + 1 =
pe ,
obtained by reversing their decomposition based on internal symmetries.
Note also that the POSet structure can be used to grade the Lie module of primes in
various ways, making gP a graded Lie algebra, prone to a direct application of Milner-Moore
Theorem (for graded bialgebras, the antipode comes for free), and making contact with the
theory of deformations and quantum groups (graded Hopf algebras) [13, 11, 12].
2. On Duality of Rationals, Adels and Dirichlet Series
This “hierarchy” opens a new perspective on the suspected duality between prime numbers and non-trivial zeroes of the Riemann zeta function, as an algebraic approach for the
celebrated Riemann Hypothesis.
2.1. On Duality: additive vs. multiplicative. Recall that Mellin transform is an exponentiated version of Fourier transform, which is the canonical isomorphism of the underlying
duality of algebraic quantum groups [19].
In more detail, Pontjagin duality of the additive group on integers can be expressed as
the short exact sequence of abelian groups
(Z, +) → (R, +) → T1
with T the 1D-torus (the circle). It is an instance of a discrete-compact duality of groups.
At the level of function spaces, we have the duality of Hopf algebras kZ and C0 (R .
Here we do not include the subtle points which are well explained in [19]. The main idea
is that extending the concept of Hopf algebra by allowing multipliers when tensoring the
duals, one obtains a nice duality theory of the so called algebraic quantum groups.
In particular for the discrete group of multiplicative rationals G = (Q×
+ , ·) [21].
2.2. Adels. On the other hand the multiplicative dual of the rationals is an infinite dimantional torus, with the group of adeles as an extension:
Q → QA → Q̂.
10
The “piece” which does not fit into the algebraic approach is that of the real numbers, result
of the completion with respect to the “infinity norm” | ± n| = n , which together with the
p-adic norms satisfy the “cocycle condition”:
Y
| |∞ ·
| |p = 1.
Let us call the “amended” duality, obtained by removing the real parts from the group
extension, the multiplicative Pontrjagin duality (to be investigated as part of the research.
It is probably expressible in terms of profinite groups.
2.3. Dirichlet Series. Now consider the duality at the level of algebraic quantum groups
[19]. We expect that Dirichlet series of arithmetic functions:
∞
X
X
DS(f )(s) =
f (n) · n−s =
< f, χ >
1
n∈(N,·)
extended the from its “Laplace form” to the whole multiplicative group Q×
+ , to play the
role of the associated canonical isomorphism.
The usual problems arrising when passing from analytic functions to Laurent series should
be dealt with. In fact it seems that the multiplicative duality obtained is a discrete analog
of Cauchy duality of complex analysis, which pairs paths and analytic functions ( T1 equivariant tangent maps). Although imprecise at this stage, the connection is clearly
prompted by Peron’s formula which uses Cauchy formula, and which provides an inverse to
Mellin transform. The conceptual connections must be further investigated.
2.4. The Riemann Zeta Function and the POSet of Primes. It is defined as the
Dirichlet series of f = 1 . In the theory of Laplace transform thsi corresponds at the level
of distributions to the delta function, as a convolution unit. When considering Fourier
Transform, we would have to think of it as represented by a sequence of approximations,
interpretation useful when using the duality theory of algebraic quantum groups, where the
convolution algebra of finitely supported functions kf in G is in duality with the group ring
kG (AQG).
The role of the POSet of prime numbers, defining the homological side of a path integral,
“a la” Feynman (with some unknown propagator), seems to play a promising role in defining
a pairing “a la” Cauchy, with respect to arithmetic functions (to be clarified by the proposed
research).
Since the multiplicative rationals (Q×
+ , ·) is just a direct sum of Z , or better a function
space:
Y
n=
pep ↔ e : P → N,
p∈P
it is interesting to investigate its extension as a dual of the Lie module of the prime numbers:
gC = Cf in P,
∗
gC
3 e : gC → C,
especially in the context of extending the characters nm to complex functions ns .
It is well known that the zeta value ζ(2) is a “propagator” for primon model, representing
the probability for two natural numbers to be relatively prime (not correlated / “noninteracting” in the lattice) [?].
It is also known that the values of the zeta function restricted to the critical line s =
1/2 + it , can be interpreted as scatering amplitudes, i.e. can be obtained via the Feynman
path integral formalism.
11
Pondering on the above connections is expected to advance the understanding of the
duality between the prime numbers, as levels of energy of a Lagrangian quantum system in
the feynmann formalism (homological algebra duality), and non-trivial zeros of the Riemann
zeta function, or better, the poles of the fermionic zeta function:
−
ζ (s) =
∞
X
µ(n) · n−s ,
ζ − · ζ = 1,
1
as resonances or periods of the POSet relative to a suitable propagator.
A natural candidate for such a propagator could be associated with the lattice structure:
corr(q, q 0) = p,
tp = ∩(tq , t0q ),
as the “common inner structure” of the two primes; e.g. 31 = 2(3·5)+1 and 43 = 2(3·7)+1
have 3 as a common “piece of structure”.
3. Dirichlet Series, Melin Transform and The Elliptic Curves Paradigm
Recall that group duality induces a Hopf algebra duality at the level of the group ring
and functions on the group:
Z
Z → R → T,
: C(T) × T̂ → C.
The LHS is Pontrjagin duality, while the RHS is the perfect pairing (“duality”) which
induces the Fourier transform as a canonical isomorphism.
We will focus on the ideas for simplicity, avoiding the technical spects.
On the “continuous side”, Mellin Transform is the exponential version of Fourier Transform, through the group isomorphism exp : (R, +) → R+ ), ·) . On the other “discrete side”
(see Pontrjagin duality), “additive number theory” and Multiplicative Number Theory are
similarly connected, with Dirichlet Series as the multiplicative group analog, plying the role
of the Fourier isomorphism [20].
The later (FT) extends to a nice Hilbert space formalism isomorphism on L2 -spaces, with
the trade-mark signature of Plancherel Theorem. This is equivalent to Parseval Theorem
(continuum / discrete):
Z
Z
Z
X
2
ˆ
f (x)|2 dx =
fˆ(ξ)dξ,
|f(n)|
=
f (t)|2 dt.
R
R
T
The Fourier Transform fˆ corresponds to the self-dual group (R, +) .
The extension for the discrete side is done using Cauchy integral, which can be viewed
as a homological / homotopical pairing (duality), allowing to extend the Riemann Zeta
function ζ(s) for instance to a meromorphic function.
Now the exponential relating the complex versions of these transforms, is the notable
main ingredient in the Theory of Elliptic Functions:
exp : (C, +) → (C× , ·),
z 7→ e2πiz ,
which can be viewed as a Riemann surface.
It allows to parametrize an Elliptic Curves as a torus: the Abel-Jacobi map (see [23] for
details, p.47-79). It is the 2D-trigonometric parametrization of an elliptic curve, with its
two components, the Weierstrass function and its derivative (analog of the 1D-trigonometry
with “ sin ” and “ cos ”).
12
Now the formalism caries over at a conceptual level to the theory of Dirichlet Series as
meromorphic functions, e.g. the “Weierstrass form” of the Riemann zeta function ([1], p.52;
sign changed for emphesis of path integral formalism interpretation):
X 1
ζ 0 (s) ζ 0 (0)
1 s X 1 s
−
=[
]0 +
|0 −
|s0 ,
ζ(s)
ζ(0)
1−x
ρ
−
s
x
+
2n
ρ
where the lattice is N with its “dual”, the set of non-trivial zeroes of the Riemann zeta
function, conjectured to lie on the critical line s = 1/2 + 2πit .
The formalism for Elliptic Functions ( g = 1 Riemann surfaces) of Weierstrass / jacobi
elliptic functions, theta-functions etc. clearly applies here, somehow. Thinking of the
Dirichlet Series in Euler product form, and the porresponding factors of p-series as p-adic
numbers (except f (k) need not be a p-adic digit):
YX
DS(f )(s) =
f (k)p−ks ,
p∈P
we see how the Dirichlet Series “lands” in an infinite product of tori, one for each prime p .
The proposed research will study this idea, its relation with the adelic duality, and if the
Dirichlet Series is a sort of a super-position of tori (Elliptic Curves), in a way similar to the
Fourier Series of 1D-trigonometry of the circle.
It is important to understand that the condition of holomorphy, at tangent map level, is
an T1 -equivariance condition. So, here we deal with 2D-equivariance cohomology (theta
functions and complex bundles).
The main novel question at this point is
Question: What is the role of the partial order on primes, i.e. do the automorphism
“content” of finite fields occur as weights in the duality, via some generating function?
Again, the convolution structure of the POSet of primes, should be related to the multiplicative Dirichlet convolution algebra structure, via a path integral formalism.
There are a few additional clues in this direction, of exploiting duality of characters. If
s
the (algebraic) characters of (Q×
+ , ·) are χs (n) = n , the the Riemann Hypothesis can be
interpreted as a C∗ -algebra unitarity condition relative to a “Pontrjagin” multiplicative
duality:
χχ † (n) = Id(n), χ†n (s) = ns̄ ,
since if s = σ + 2πit then this amounts to:
nσ+2πit · nσ−2πit = n2σ ∼
= n ↔ 2σ = 1,
i.e. σ = 1/2 ( s is on the critical line).
3.1. Riemann-Mangoldt Formula and Parceval Theorem. We also note, an passant,
that the duality interpretation of Dirichlet Series explained above leads to another claim,
that the Riemann-Mangoldt Formula [1], p.54 (rewritten for our purposes):
X xρn x−2n
x − ψ(x) = +
−
,
ρn
n∈N
is “just” Parceval’s Theorem (with maybe a “central charge”?). Here ψ(x) is Cebishev’s
function, essentially:
X
ψ(x) =
log p,
pn <x
13
in terms of which the Prime Number Theorem is expressed as ψ(x) ∼ x ,
Then, with a higher harmonic analysis flavor in mind (from 2D-trig to ∞ -trig”), we
claim that the “Weierstrass form” of the Riemann zeta function (or rather of d/ds log ζ )
can be interpreted as a superposition of charges corresponding to a Poisson equation for
the theta function σ(s) [23], p.69 (not Re(s) , in disguise:
d
ζ(s) =
log σ(s), −ζ 0 (s) = P(s).
ds
The equations which have to be better understood is the “Poisson Equation” for the inverse
of the Abel-Jacobi isomorphism α(z) [23], p.63:
2
∂
log σ(z) = α−1 (z).
∂z
Recall that the Abel-Jacobi map is the path integral (a potential function / “logarithm”):
Z P
α : E(C) → C/L, α(P ) =
ω(mod L).
O
Relevant for the Riemann Hypothesis is the change of variables from the torus (Elliptic
Curve in angular coordinates) to the multiplicative torus (additive to multiplicative groups):
C/(Z + Zτ ) → C× /q Z ,
where the lattice determining the Riemann surface structure is L = Z + Zτ .
Therefore it is natural to ask “What is the role of the Critical Line pulled-back to the
lattice?”
3.2. Primon Model and Breaking the Symmetry. Returning to the physics interpretation, the “charges” are the zeroes of ζ (poles of the fermionic zeta function ζ − = DS(µ) ).
The energy levels are the logp ; the associated probabilities are p , the prime numbers. The
POSet opens the idea that there are “jumps” corresponding to breaking the symmetry, from
p to q if q|p − 1 .
Recall that the zeta zeroes “break” the Argand diagram graph of the zeta function
restricted to the critical line into loops. The interpretation of zeta values as scatering
amplitudes should be relevent at this point.
4. Riemann-Siegel formula and Gutzwiller trace formula
The technique for locating the (non-trivial) zeroes of the zeta function, due to Riemann,
as recovered by Siegel from his notes, makes use of the following decomposition:
1
1
it
ζ(1/2 + it) = e−θ(t) Z(t), θ(t) = Im{log Π( (−1 − [ − it])) − Π(−1/2)},
2
2
2
which seems to “remove” some (the?) gamma function contribution.
The Riemann-Siegel formula is a technique for the approximate numerical evaluation of
the remainder in integral form, which yields an approximation of Z(t) [1], p.139:
Z(t) ∼ 2
N
X
n−1/2 cos[θ(t) − t log n].
1
The Gutzwiller trace formula [?], p.218, with an obvious similar form as the above RiemanSiegel approximation, is an asymptotic Fourier decomposition of single-particle states in
terms of classical periodic orbits [?], p.207.
14
The main points are: 1) Multiplicative vs. additive number theory is an analog of
quantum vs. semi-classical theory of dynamics of bound states; 2) The corresponding
dualities are represented by Dirichlet vs. Fourier transforms; 3) The analog of “resonant
states” in physics should correspond to symmetry correlations between prime numbers
(automorphisms of basic finite fields), and reflected numerically in the POSet structure.
The sums involved in both formulas are reminiscent of Gauss sums (quantum Fourier
transform), and Ramanujan-Fourier sums [?]. Coincidentely, the correlation between primes
using using the Wiener-Khintchine machinery was studied for prime pairs (p − h/2, p + h/2)
[?] via Ramanujan-Fourier series for arithmetic functions.
Now the zeta function seems to play the role of a fundamental solution (“Green function” propagator), as the Dirichlet transform (series) of the constant function (sum of all
characters).
The proposed research will aim to understand in more depth the above related facts and
concepts.
5. Plan of Research and Development
There will be three parallel activities, yet connected via feedback.
1) (Research) The first main task of the PI is to clarify the above statements, and
formulate precise statements following the suggestions inserted in the GP.
2) (Feed-back) Next, the PI intends to give a series of presentations at the nearby mathematics departments (Urbana-Champagne and Northwestern), to receive feedback, collect
suggestions from mathematicians more versed in the areas of Number Theory and Algebraic
geometry. On this occasions, the weak links of the approach will be also identified; and
then fortified. explaining these ideas several times is the best way to clarify them at the
level of concepts and the corresponding big picture.
3) (Personal development) Further learning the Theory of Modular Forms, and additonal
techniques of algebraic-geometry as needed.
The PI is a life-long learner. For a better understanding of the topics mentioned in this
GP, and learned over the past few years, the PI taught several times courses on Advanced
Number Theory and Advanced Topics in Mathematics [?], and also presented in the Algebra
Seminar, consolidating the main ideas involved, and developing the theory of the internal
structure of the prime numbers (finite fields) [?].
4) (Research and Dissemination of Results) The “rest” (99%) is hard research work, while
routinely publishing the findings.
The disemination of the associated research questions is equally important, since the
new avenue of research in Number Theory originating with the intyernal structure of prime
numbers is a vast unexplored territory, as far as the PI could tell; and priority of results is,
in PI’s opinion, totally secondary, especially when dealing with such difficult a subject.
6. Conclusions and Further Developments
6.1. Multiplicative Number Theory as Ultimate Physics Theory. Once upon a
time (in college), the PI encountered the reference [6]; that physics is math, seemed a nice
day-dream for mathematicians ... and then, efortlessly, Kontsevich’s Formality Theorem,
Connes-Kreimer renormalization, and the new Quantum Computing paradigm, led the way
naturally, to the unavoidable conclusion: the “World” is digital, but the “unit” is the qubit,
which under measurement (gauging) becomes the bit: True/False, Up/Down, Warm/Cold
etc. It is the elementary building block, yet not a “particle” because although localized, it
15
is also a diuality (“wave aspect”). It is not just a fermion, because it also can play the role
of a quantum gate, i.e. a unit of interaction (boson).
In this flow on networks oriented approach, the basic modes of “vibration” (bound quantum states) of systems in general (e.g. Hydrogen atom etc.) have to be reducible to prime
numbers (basic finite fields/ cycles/ periods), when modeled by algebraic simple objects,
like in a categorical approach to conformal field theory [?]. Another “clue” in this direction
of thought, is the use of Euler’s integral in the definition of Veneziano amplitude, satisfying
a relation similar to the one satisfied by the gamma function.
Such a line of reasoning opens a way for explaining the fundamental constants in terms
of Number Theory. Several “clues” were discovered by the PI and reported in a “crude”
form in [2]. There are various other contributions along these lines [?, 7] etc.
A progress in this direction seems to depend at this stage on the understanding of primes,
Riemann zeta function as the core of an algebraic quantum group duality, and then to
connect with physics via the H-atom spectrum, as the obvious candidate for a physics
analog of a qubit.
6.2. Understanding Multiplicative Number Theory First! Returning to the mathematics side, the present GP aims to study the internal structure of finite fields as shadows
of primes, and its implications in the duality which lies at the core of Ruemann Hypothesis.
The starting point is PI’s discovery of the partial order on the set of prime numbers.
It is a natural consequence of the interpretation of a finite field as a Klein geometry, and
therefore considering the orbits of the automorphisms of the finite field. In other words, we
“decompose” the prime according to its symmetries.
It is a nice surprise that the prime analogues of quarks, the Fermat primes conjectured
to be five, together with p = 2 , is in a one-to-one correspondence with quarks, which
approximately preserves the logaithm of their masses. This is compatible with a probabilistic
interpretation of the primes, within the Riemann gas model, and also consistent with the
other “coincidence” which regards the approximation of the logarithm of the fine structure
constant by the prime zeta function value at s = 2 (it can also be interpreted as a “total
L2 -norm”, like an L2 -partition function of the basis of prime numbers.
There are rich consequences and relationships between the POSet of primes, the Hopf
algebra of rooted trees, within the framework of algebraic quantum groups. The theory
of Lie groups, adapted to modules fits perfectly as a tool of study, while the duality of
multiplicative number theory having Dirichlet series as an analog of the canonical Fourier
isomorphism, seems to invite to consider a modification of the classical duality of rational numbers, which involves the field of adeles, as a candidate for the duality behind the
Riemann Hypothesis, towards the explanation of the relationship between prime numbers
(basis of the Lie module of rationals) and the non-trivial zeroes of the Riemann zeta function
(“charges” determining the meromorphic zeta function).
The various questions and ramifications of the topics presented, constitute topics which
need be investigated, forming the core of the proposed research, as it appears to the PI at
this stage.
Furthermore, the top-down/ conceptual investigation of the PI will help systematize the
problematic, and open new paths of investigation in this new mathematics teritory centered
on the internal, algebraic-geometric structure of the prime numbers.
1
On Prime Numbers and Riemann Hypothesis
by L. M. Ionescu
References
[1] H. M. Edwards, Riemann’s Zeta Function, Dover Publications, 1974.
[2] L. M. Ionescu, Remarks on Physics as Number Theory, http://my.ilstu.edu/ lmiones/309981.pdf, 2011.
[3] Proth Primes, http://primes.utm.edu/glossary/xpage/ProthPrime.html, http://www.prothsearch.net/
etc.
[4] Robert Baillie, New Primes of the Form k · 2n + 1 , Mathematics of Computation, Vol.33, No.148, )ct.
1979, pp.1333-1336.
[5] G.V. Cormack, H.C. Williams, Some very large primes of the form k · 2m + 1 , Mathematics of Computation, Vol. 35, No. 152, Oct. 1980, Pp. 1419-1421.
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0-9538552-0-1.
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M.
Ionescu,
The
Rationals
as
an
Algebraic
Quantum
Group,
http://my.ilstu.edu/ lmiones/presentations drafts.htm, Nov. 15, 2011.
[22] J. H. Silverman, A Friendly introduction to number theory, Prentice Hall, 2001.
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