Towards a statistical proof of the Riemann Hypothesis

Abstract

Using the $\zeta$ functional equation and the Hadamard product, an analytical expression for the sum of the reciprocal of the $\zeta$ zeros is established. We then demonstrate that on the critical line, $|\zeta|$ is convex, and that in the region $0 < \Re(s) \leq 0.5$, $|\zeta|$ has a negative slope, given the RH. In each case, analytical formulae are established, and numerical examples are presented to validate these formulae.