Some simplest integral equalities equivalent to the Riemann hypothesis

Abstract

UDC 511.3

We show that the following integral equalities are equivalent to the Riemann hypothesis for any real $a>0$ and any real $0<\epsilon<1,$ $\epsilon \neq 1$: $$\begin{gather*}\int\limits_{-\infty}^{\infty}\frac{\ln\left(\zeta\left(\dfrac{1}{2}+it\right)\right)}{a+it}\,dt=-2\pi\ln\frac{a+\dfrac{1}{2}}{a}, \\ \int\limits_{-\infty}^{\infty}\frac{\ln\left(\zeta\left(\dfrac{1}{2}+it\right)\right)}{(a+it)^\epsilon}\,dt=-\frac{2\pi}{1-\epsilon}\left(\left(a+\frac{1}{2}\right)^{1-\epsilon}-a^{1-\epsilon}\right).\end{gather*}$$