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: