Towards a statistical proof of the Riemann Hypothesis

TL;DR

The paper develops analytical expressions related to the Riemann zeta function, demonstrating properties of its magnitude on the critical line and in certain regions, aiming to contribute towards a proof of the Riemann Hypothesis.

Contribution

It introduces new analytical formulas for the sum over zeta zeros and properties of ||, providing numerical validation towards the Riemann Hypothesis.

Findings

|| is convex on the critical line.

|| has a negative slope in certain regions assuming RH.

Numerical examples support the analytical formulas.

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.