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Distributional PNT

periodically updated until February 2017

There is only a formal representation of the Zeta function as transform of a Gaussian function based operator ((EdH) 10.3). The operator has no Mellin transform at all as the integrals do not converge due to the not vanishing constant Fourier term of the Gaussian.

The Hilbert transformation of the Gaussian has a vanishing constant Fourier term. It is given by the Dawson function. We propose an alternatively Zeta function theory based on the Mellin transform of this function showing same singularity behavior (s=0,1) as the Zeta function (in contrast to the Gamma function, which is the Mellin transform of the Gaussian function). The Dawson function asymptotics provides appreciated convergence behavior (in contrast to the Gaussian function) overcoming current related RH criteria challenges.

The same concept can be applied to the fractional part function resp. its Hilbert transform.

The corresponding alternative asymptotic density functions are considered:

Braun K., A Kummer function based Zeta function theory to prove the Riemann Hypothesis and the Goldbach conjecture, 28.2.2017

Braun K., A Kummer function based Zeta function theory to prove the Riemann Hypothesis, 31.1.2017

Change history

February 2017: extension: a Kummer function based circle method to solve the binary Goldbach conjecture; a solution concept to prove the irrationality of the Euler constant

January 2017: "Summary" and "§2" revised, §2 split into §2,§3,§4; "Notes O71/72", new, "Notes S48-55", new

December 2016: "Notes O53-O70 (Yukawa potential, plasma dispersion function = Dawson function, Landau damping, reduced Hilbert transform, related Schrödinger (commutator) differential operator properties)", new

November 2016: "Summary" update; "Notes S36-47, Note O52" new

October 2016: "Summary": new;  "Note S19" updated;  "Notes S29-35, O50-51", new

August 2015: "original version"

RH 2015-2016 removed page text:


RH 2015 - 2016, page text

Further related papers

Braun K., A distributional way to prove the Goldbach conjecture leveraging the circle method

Braun K., The Goldbach conjecture, a solution concept enabled by the alternative Zeta function theory

Braun K., An alternative trigonometric integral representation of the Zeta function on the critical line

In a nutshell, The solution concept to prove the RH

An alternative framework to answer the RH

Braun K., Hilbert transformation, Gaussian and Dawson function, and the Harmonic Quantum Oscillator Model

Braun K.,Some commutator properties

Braun K.,The Prandtl (hyper singular integral) operator with double layer potential

Braun K. Generalized wavelet theory and non-linear, non-periodic boundary value Problems

Braun K., Infinite continued fraction representations of the exponential integral function, Bessel functions and Lommel polynomials

Braun K., Lommel and Hermite polynomials

Braun K., Some remarkable Pseudo-Differential Operators of order -1, 0, 1

Theorems from Polya, Muentz, Ikehara, Wiener, Ramanujan, Nyman, Theodorsen, Frullani and Hardy

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