A version of Carleman's formula and summation of the Riemann $zeta$-function logarithm on the critical line

A version of Carleman's formula for functions holomorphic in a rectangle is proved. It is applied to the evaluation of the integral of $zeta$-function logarithm with the summing factor $exp(-t)$ along the critical line. This allowed to obtain a new statement equivalent to the Riemann hypothesis.