The validity of the analog of the Riemann hypothesis for some parts of $\zeta(s)$ and the new formula for $\pi(x)$

TL;DR

This paper proves an analog of the Riemann hypothesis, introduces new integral equations for prime counting functions, and reveals that these functions are solutions to similar equations despite their different behaviors.

Contribution

It establishes an analog of the Riemann hypothesis and derives new integral equations for functions related to prime counting, showing their solutions share similar properties.

Findings

Proved an analog of the Riemann hypothesis.

Derived new integral equations for $oldsymbol{oldsymbol{ ext{ extpi}}(x)}$ and $oldsymbol{ ext{R}}(x)$.

Showed these functions are solutions to similar integral equations despite different behaviors.

Abstract

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}.