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 this function. Corresponding confluent hypergeometric functions with related singularity behaviors at the critical line and at s=0,1 enable appreciated convergence behavior of the Zeta function to apply existing RH criteria. Here we are (original version, August 2015) August 2016 update: additional appendix section with "opportunity notes", e.g. notes O24-O30) avoiding Weyl sums estimates to solve the Goldbach conjecture
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