On an equality equivalent to the Riemann hypothesis - Volchkov, V. V. (Ukrains’kyi Matematychnyi Zhurnal, 1995)
We prove that the Riemann hypothesis on zeros of the zeta function ζ(s) is equivalent to the equality $$\int\limits_0^∞ {\frac{{1 - 12t^2 }}{{(1 + 4t^2 )^3 }}dt} \int\limits_{1/2}^∞ {\ln |\varsigma (σ + it)|dσ = π \frac{{3 - γ }}{{32}},}$$ where $$γ = \mathop {\lim }\limits_{N → ∞ } \left( {\sum\limits_{n = 1}^N {\frac{1}{n} - \ln N} } \right)$$ is the Euler constant.