A Hidden Symmetry Related to the Riemann Hypothesis with the Primes into the Critical Strip

TL;DR

This paper explores a novel symmetry related to the Riemann Hypothesis by analyzing integrals of the logarithm of the Zeta function with primes, proposing a new formulation and numerical evidence supporting RH.

Contribution

It introduces a new equivalent formulation of the Riemann Hypothesis based on a hidden symmetry involving the critical strip and primes.

Findings

Formulates RH as a hidden symmetry connecting inside and outside the critical strip.

Provides a numerical experiment supporting the plausibility of RH.

Extends previous integral treatments with generalized Lorentz measures.

Abstract

In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We first obtain two new equivalent formulations of the Riemann Hypothesis (RH). Then with a special choice of the measure we formulate the RH as a ``hidden symmetry'', a global symmetry which connects the region outside the critical strip with that inside the critical strip. The Zeta function with all the primes appears as argument of the Zeta function in the critical strip. We then illustrate the treatment by a simple numerical experiment. The representation we obtain go a little more in the direction to believe that RH may eventually be true.