A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis III. Exponential weight functions

TL;DR

This paper introduces new integral equalities involving the logarithm of the Riemann zeta-function with exponential weights, some of which are numerically verified and shown to be equivalent to the Riemann hypothesis.

Contribution

It establishes novel integral equalities with exponential weight functions, including those along the real axis, and provides numerical and rigorous estimates related to the Riemann hypothesis.

Findings

Numerical verification of an integral equality involving log|zeta(1/2+it)| and cosh(pi*t)^(-1) up to 80 digits.

New integral equalities involving exponential weights like exp(-at) are introduced.

Zero contributions off the critical line are shown to be negligibly small, less than 10^(-10^(13)) for a=4*pi).

Abstract

This paper is a continuation of our recent papers with the same title, arXiv:0806.1596v1 [math.NT], arXiv:0904.1277v1 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, which this time involve exponential functions, are established, and for the first time we have found equalities involving the integrals of the logarithm of the Riemann zeta-function taken exclusively along the real axis. Some of the equalities we have found are tested numerically. In particular, an integral equality involving the logarithm of abs(zeta(1/2+it)) and a weight function cosh(pi*t)^(-1) is shown numerically to be correct up to the 80 digits. For exponential weight function exp(-at), the possible contribution of the Riemann function zeroes non-lying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than trillion of digits, 10^(-10^(13)), for a=4*pi.