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

Abstract

This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type are established; contrary to the preceding paper the focus now is on integrals involving the argument of the Riemann zeta-function (imaginary part of logarithm) rather than the logarithm of its module (real part of logarithm). Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented. Our integral equalities, together with the equalities given in the previous paper, include all earlier known criteria of this kind, viz. Wang, Volchkov and Balazard-Saias-Yor criteria, which are certain particular cases of the general approach proposed.

Cited by in corpus (4)

• First applications of generalized Li's criterion to study the Riemann zeta-function zeroes location
• Analysis of Voros criterion: what derivatives involving the logarithm of the Riemann xi-function at z=1/2 should be non-negative for the Riemann hypothesis holds true
• Asymptotic of the generalized Li's sums which non-negativity is equivalent to the Riemann Hypothesis
• Errata and Addenda to Mathematical Constants