A few equalities involving integrals of the logarithm of the Riemann Zeta-function and equivalent to the Riemann hypothesis

TL;DR

This paper introduces new integral equalities involving the Riemann zeta-function, rigorously proving their equivalence to the Riemann hypothesis and providing preliminary numerical tests.

Contribution

It establishes novel integral equalities linked to the Riemann hypothesis using a generalized Littlewood theorem, with separate analysis of real and imaginary parts.

Findings

New integral equalities equivalent to the Riemann hypothesis

Rigorous proof of equivalence to the Riemann hypothesis

Preliminary numerical tests supporting the equalities

Abstract

Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functions, we have established a few equalities involving integrals of the logarithm of the Riemann Zeta-function and have rigorously proven that they are equivalent to the Riemann hypothesis. Separate consideration for imaginary and real parts of these equalities, which deal correspondingly with the integrals of the logarithm of the module of the Riemann function and with the integrals of its argument is given. Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented.