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:

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