The validity of the analog of the Riemann hypothesis for some parts of $ζ(s)$ and the new formula for $π(x)$

An analog of the Riemann hypothesis is proved in this paper. Some new integral equations for the functions $\pi(x)$ and $R(x)$ follows. A new effect that is shown is that these function - with essentially different behavior - are the solutions of the similar integral equations. \noindent This paper is the English version of the paper of reference \cite{1}.