On an equality equivalent to the Riemann hypothesis

Abstract

We prove that the Riemann hypothesis on zeros of the zeta function ζ(s) is equivalent to the equality

$$\int\limits_0^\infty {\frac{{1 - 12t^2 }}{{(1 + 4t^2 )^3 }}dt} \int\limits_{1/2}^\infty {\ln |\varsigma (\sigma + it)|d\sigma = \pi \frac{{3 - \gamma }}{{32}},}$$

where

$$\gamma = \mathop {\lim }\limits_{N \to \infty } \left( {\sum\limits_{n = 1}^N {\frac{1}{n} - \ln N} } \right)$$

is the Euler constant.