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                               www.riemann-hypothesis.de
                               www.goldbach-conjecture.de


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 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-Vandiver 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 15, 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