Riemann hypothesis and some new integrals connected with the integral negativity of the remainder in the formula for the prime-counting function $\pi(x)$

TL;DR

This paper introduces a new integral representation for the remainder in the prime-counting function and proves the existence of infinitely many related formulas, advancing understanding of prime distribution.

Contribution

It presents a novel integral for the prime-counting remainder and establishes an infinite set of formulas involving this integral, linking to the Riemann hypothesis.

Findings

New integral formula for the prime-counting remainder

Existence of infinitely many formulas involving the integral

Connections to the Riemann hypothesis

Abstract

In this paper a new integral for the remainder of $\pi(x)$ is obtained. It is proved that there is an infinite set of the formulae containing miscellaneous parts of this integral.