The RH solution enabled by a Kummer functions based Zeta function theory
The Riemann Hypothesis states that the non-trivial zeros of the Zeta function all have real part one-half. The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Zeta function corresponds to eigenvalues of an unbounded self-adjoint operator.
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.
Essentially we propose a three pillar concept based on Hilbert scale, Hilbert transforms and Hilbert-Polya conjecture, which we therefore call "triple H" concept. The Bagchi criterion is somehow the Hilbert-Polya conjecture based on the fractional part function and its related Hilbert transform.
Additionally we propose a five pillar based concept to overcome current challenges to solve the binary Goldbach conjecture.
The “triple H” concept can also be applied to existing challenges in theoretical physics models, e.g. Bose-Einstein statistics and the related Planck black body radiation law, Boltzman statistics, Yukawan potential theory, magnetized Bose plasma, Boltzman equation and Landau damping, non-local transport theory (Notes O52 ff. and sections NSE, YME, GSE).