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Alternatively to the current Gaussian function based zeta function theory a Kummer function based zeta function theory is proposed to enable

      - the verification of several RH criteria
      - a proof of the binary Goldbach conjecture.


A. The Riemann Hypothesis and related conjectures 

1. The Riemann Hypothesis 

The Riemann Hypothesis states that all non-trivial zeros of the zeta function have real-part one-half.  The conceptual linkage to the UFT pages is given by the Hilbert-Polya conjecture and the Berry-Keating conjecture  

2. The Hilbert-Polya conjecture 

The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Zeta function corresponds to eigenvalues of an unbounded self-adjoint operator.  

3. The Berry-Keating conjecture 

(DeJ) p. 315: „The Berry-Keating conjecture is about an unknown quantization H of the classical Hamiltonian H=xp, that the Riemann zeros coincide with the spectrum of the operator ½+iH. This is in contrast to the canonical quantization, which leads to the Heisenberg uncertainty principle and the natural numbers as spectrum of the harmonic quantum oscillator. The Hamiltonian needs to self-adjoint so that the quantization can be a realization of the Hilbert-Polya conjecture.“ 

4. The Goldbach conjecture 

The binary Goldbach conjecture states that every positive even number n > 2 is the sum of two primes. 

5. The Kummer conjecture 

The Kummer conjecture is about the distribution of the cubic exponential sums with p = 1 mod 3. It is intimitely connected to the real part of the normalized cubic Gauss sums over Eisenstein integers, (DuA), (KuE6). It is the real part of an explicit root of unity with range (-1,1). The observed „Kummer ratio“ between the "range" interval split into (-1, -1/2), (-1/2,1/2), (1/2,1) was 3 : 2 : 1. 

        

Braun, K., The Kummer conjecture and the two-semicircle method

                                                 July 30, 2022


6. The Courant conjecture

(CoR) p. 763: Families of spherical waves for arbitrary time-like lines exist only in the case of two or four variables, and then only if the differential equation is equivalent to the wave equation.


B. The distribution of prime numbers and the behavior of subatomic particles

(DeJ)p. 295: „The non-trivial zeros of the Riemann Zeta function arise from inquiries into the distribution of prime numbers. The eigenvalues of a random Hermitian matrix arise from inquiries into the behavior of systems of subatomic particles under the laws of quantum mechanics. What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?“


C. The Riemann Hypothesis and the zeta function 

1. The connection between the zeta function and the primes 

The Euler product formula is an application of the sieve of Eratosthenes to the Riemann Zeta function for Re(s)>1. The Dirichlet series of zeta(s) defined for Re(s)>1 (running through all the positive whole numbers n) is equal to Euler’s (infinite) product formula (running through all prime numbers). This is sometimes called the „golden key“, (DeJ) 7.2. Turning this key is enabled by Riemann’s density function J(x); its the Mellin transform is given by log(zeta(s))/s for Re(s)>1, (EdH) 1.12. The combination of the latter term with the Riemann duality equation then gives Riemann‘ s famous formula for J(x); the principle term of J(x) is derived from the term -log(s-1) leading to the li(x) density function, (EdH) 1.14. 

2. The entire Zeta function and the Riemann duality equation 

The Riemann duality equation is given by Z(s)=Z(1-s) with an appropriately defined entire Zeta function Z(s). One proof of the Riemann functional equation of the Zeta function evaluates first the contour integral for negative real values, (EdH) 1.6. The extention for all s (except s=0,1,2,.. where one or more of the Terms of  the functional equation have poles) is based on the relationship of the zeta function at the values (2n-1) and the Bernoulli numbers, (EdH) 1.6. Another proof of the functional equation is based on the equivalent functional equation of Jacobi’s theta function, which is basically the Poisson summation formula for the Gaussian function, (EdH) 1.7.

The Riemann duality equation is given by Z(s)=Z(1-s) with an appropriately defined entire Zeta function Z(s). One proof of the Riemann functional equation of the Zeta function evaluates first the contour integral of zeta(s) for negative real values, (EdH) 1.6. The extention for all s (except s=0,1,2,.. where one or more of the Terms of  the functional equation have poles) is based on the relationship of the zeta function at the values (2n-1) and the Bernoulli numbers, (EdH) 1.6. Another proof of the functional equation is based on the equivalent functional equation of Jacobi’s theta function, which is the Poisson summation formula for the Gaussian function, (EdH) 1.7. 

3. The Dirichlet series and the (contour) integral representation of the zeta function

For Re(s)>1,Riemann’s integral formula representation of the Dirichlet series representation of the zeta function is built on the Mellin transform of the Poisson sum of the Gaussian function (resp., by variable substitution, of the Poisson sum of the exponential function for x<0). Riemann’s contour integral representation of the zeta function (primarily defined for Re(s)<1)) is analytic at all points of the complex s-plane except for a simple pole at s=1.This function coincides with the Dirichlet series representation of the Zeta function for Re(s)>1, (EdH) 1.4.


D. Conceptual challenges

1. The Gaussian function based zeta function theory

Riemann's work to develop this the theory is based on Fourier analysis. All known integral representations of the entire Zeta function Z(s) are Fourier integrals defined in the classical "bounded" function metric space framework. The natural form of representation to prove A2. and A3. are convolutions integral. In quantum mechanics those integrals are accompanied by the concept of self-adjoint operators with corresponding Hilbert space domain framework. In other words, there is no common framework enabling a common domain definition for the operators A1. and A2.

The not vanishing constant Fourier term of the Gaussian function in the Poisson summation Formula (which is equivalent to the functional equation of the entire Zeta function) is the root of evil to build a self-adjoint operator with transform Zeta(s), (EdH) 10.3. At the same time the spectrum of a hermitian operator, whose inverse operator is not compact, is not purely discrete.

2. The Hardy-Littlewood circle method

Applying the Hardy-Littlewood circle method to prove the binary Goldbach conjecture failed due to insufficient (purely Weyl sum based, w/o any data from the problem under study) bounds for the minor arcs.  

The Schnirelmann density of the odd integers is ½, while the S-density of the even integers is zero. However, conceptually the Hardy-Littlewood circle method does not distinguish between odd and even integer.

By appealing to a heuristic form of the circle method Patterson‘s heuristic fell short of a proof of his conjecture explaining the suspected bias of the Kummer conjecture (DuA). This was also due to insufficent bounds for the minor arcs.  

We note that the Patterson conjecture is confirmed conditionally on the assumption of the Generalized Riemann Hypothesis, (DuA). We further note that there is also are refinement from the Patterson conjecture that features an error term capturing square root cancellation, (DuA). We also note that that the cubic large sieve cannot improved relying on the GRH based on a dispersion estimate for cubic Gauss sums, (DuA).


E. The proposed "common denominator" framework

All orthonormal bases of a Hilbert space H with an orthogonal basis e(n) are characterized in terms of unitary operators U acting on this orthogonal basis, U(e(n)). The spectrum of self-adjoint operators in a separable Hilbert space framework, whose inverse operator is a compact, is discrete.

The considered Kummer function are the solution of an underlying self-adjoint Whittaker partial differential equation.

The discrete spectrum in case of the Hardy-Littlewood circle method is the set of integers, which are the zeros of the orthonormal basis functions (1,sin(n*), cos(n*)).

The imaginary parts a(n) of the zeros of the considered Kummer function enjoys similar appropriate properties than the Zeros of the Digamma function. This property enables the definitions of corresponding Hilbert scales and „retarded/condensed“sequences b(n):=(3*a(n)+a(n+1))/4 with „density“ ½ fulfilling the Kadec condition, (YoR) p. 36. We mention the related Kummer "distribution" conjecture of the cubic exponential sums with p = 1 mod 3.

The definition of a Riesz basis appears by weakening the „unitary“ condition of U to bounded bijective operators, ChO)3.6. It is the central concept in the theory of non-harmonicFourier series accompanied with the concept of a Riesz basis. Therefore, the mapping (1, sin(n*),cos(n*)) to (sin(b(n)*),cos((b(n)*) enables a transfer from harmonic Fourier series analysis to non-harmonic Fourier series analysis, whereby the underlying indices domain has Schnirelmann density 1/2.

Braun K., A toolbox to solve the RH and to build a non-harmonic Fourier series based two-semicircle method

Supporting papers

                         

Braun K., Looking back, part A, (A1)-(A3)

                                             April 18, 2021

                                       

Braun K., RH solutions

                                                 July 31, 2019

Braun K., A Kummer function based Zeta function theory to prove the Riemann Hypothesis and the Goldbach conjecture

                                            March 27, 2017