Reimann Zeta Function

Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri-Lagarias, Maślanka, Coffey, Báez-Duarte, Voros and others. We apply the theory of Nörlund-Rice integrals in conjunction with the saddlepoint method and derive precise asymptotic estimates. The method extends to Dirichlet Lfunctions and our estimates appear to be partly related to earlier investigations surrounding Li's criterion for the Riemann hypothesis. √ πn .

By means of interferences between prime functions the article shows how an asymmetry between complex conjugates non-trivial zeros inside of the critical strip appears in the Riemann Zeta Function when the prime harmonic functions have a different phase, which could challenge the Riemann Hypothesis while clarifying the relation between prime numbers and the Riemann non-trivial zeros.