Riemann Hypothesis
Unified Field Theory
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A Kummer function based Zeta function theory is proposed, alternatively to the current Gaussian function based theory. Basically, this is a replacement of the Gaussian function by its Hilbert transform, which is equal to the Dawson function. It results into an alternatively defined entire Zeta-function accompanied by the product representations of the functions Gamma(1-s/2), 2a*sin(a*s) and a*tan(a*s) with a:=pi/2, where at s=1 the latter function has the same singularity as the extended meromorph zeta function on the half-plane Re(s)>0.

The corresponding alternative representation of the duality equation is accompanied by

- an alternative contour integral representation of the zeta-function for Re(s)<1, which is consistent with the Dawson function based Mellin integral transform representation of the alternative entire Zeta-function for 0<Re(s)<3/2

- the set of the imaginary parts of the only complex-valued zeros of the Dawson function enjoying similar appreciated properties as the zeros of the Digamma function.

The sum of the inverse Fourier transforms of log(2a(sin(as))) and log(a(tan(as))) in combination with two appropriately defined „Landau“ sequences with indices domains (4n-3) and (4n-1), n>0, provides an alternative number theoretical density function to li(x).

The Fourier coefficients of log(a(tan(ax))) are given by the sequence h(n)/n with h(n):=H(2n)-H(n)/2, where H(n) denote the harmonic numbers. The corresponding Dirichlet series defines a new approximating zeta-function for Re(s)>1.

The alternating indices domains (4n-3) and (4n-1), n>0, enable a newly proposed two-semicircle method, where each semicircle is governed by one of the two related density functions. It replaces the major/minor arcs division concept of the Hardy-Littlewood circle method.

The alternating indices domains (4n-3) and (4n-1), n>0, also permits a revisit of Kummer's "ideal complex number" concept based on a proposed (Euler,Kummer) = (4n-3,4n-1) "pairing" concept in a quaternionic setting.

In summary, the alternative entire Zeta function, which is represented as a Dirichlet series built on the Hilbert transform of the Gaussian function for Re(s)>1,  accompanied by a corresponding alternative integral representation of zeta(s) in the critical stripe

-      simplifies the verifications of several RH criteria

-      enables the Landau approach to prove the Goldbach conjecture

-      permits a re-examination of the Kummer conjecture.

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

                                     scope of application: pp. 2-4 

                                      May 23, 2022 update: p. 4

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                     

Further supporting data


Alpay D., et. al., Inner Product Spaces and Krein Spaces in the Quaternionic Setting

B. Bagchi On Nyman, Beurling and Baez-Duarte s Hilbert space reformulation of the RH.pdf


M.V. Berry I.P. Keating H=xp and Riemann zeros


Beurling A. A closure problem related to the Riemann Zeta-function


Burnol J.-F., A note on Nyman s equivalent formulation of the Riemann Hypothesis


Burnol Scattering, determinants, hyperfunctions in relation to Gamma(1-s)Gamma(s).pdf


Cardon Convolution operators and zeros of entire functions


Edwards H. M., Riemann s Zeta Function


Ginzel I., Die konforme Abbildung durch die Gammafunktion


Kac M., Statistical Independence in Probability, Analysis and Number Theory


Kac M., Probability methods in some problems of analysis and number Theory

Kummer E. E., Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren


Landau E., Ueber eine trigonometrische Reihe


Landau E., Ueber die Fareyreihe und die Riemannsche Vermutung


Landau E., die Goldbachsche Vermutung und der Schnirelmannsche Satz


Landau E., Ueber die zahlentheoretische Function und ihre Beziehung zum Goldbachschen Satz


Landau E., Handbuch der Lehre von der Verteilung der Primzahlen I


Landau E., Handbuch der Lehre von der Verteilung der Primzahlen II


(LeB) Lectures on Entire Functions, Lecture 5


(LeN) Gap and density theorems, VI and VII


Lebedev N. N., Special Functions and their Applications


LeFloch P. G., Ma Y., The global nonlinear stability of Minkowski space


Linde A., Inflation, Quantum Cosmology and the Anthropic Principle


Tong H., Introducing Quaternions to Integer Factorization


(PaR) Fourier transforms in the complex domain, 22 and 33

Polya Ueber eine neue Weise bestimmte Integrale in der Zahlentheorie zu gebrauchen


Riemann article


Riemann handwritten paper from 1859


Sedletskii A. M., On the Zeros of Laplace Transfroms


Vaaler J. D., Some extremal functions in Fourier analysis


Vindas J., Estrada R., A quick distributional way to the prime number Theorem


Wang Yuan, The Goldbach conjecture


Wiener N., The mean square modulus of a function


5 RH criteria, lecture notes