One functional property of the $\varsigma$-function of Riemann

TL;DR

This paper establishes a connection between the holomorphic extension of a specific integral function related to prime counting and the zero-free regions of the Riemann zeta function, implying a potential approach to the Riemann hypothesis.

Contribution

It proves that extending the function heta(z) holomorphically into a domain implies the Riemann zeta function has no zeros there, offering a new perspective on the Riemann hypothesis.

Findings

Holomorphic extension of heta(z) implies zero-free regions for unction.

If heta(z) is holomorphic for rac{1}{2}<Re(z), then Riemann hypothesis holds.

Provides a criterion linking integral function properties to the zeros of unction.

Abstract

We prove that if a function $\theta \left( z \right)=\int\limits_{1}^{\infty }{\frac{\pi \left( t \right)\,-Li\left( t \right)}{{{t}^{z+1}}}dt}\,,$ which is holomorphic in $\left\{ \operatorname{Re}z>1 \right\}$ holomorphically extends to some simply connected domain $G\subset \left\{ \operatorname{Re}z>\frac{1}{2} \right\}$, then the $\varsigma \left( z \right)-$function of Riemann has no zeros in this domain, $\varsigma \left( z \right)\ne 0\,\,\,\forall z\in G.$ As a consequence, it turns out that if the function $\theta \left( z \right)$is holomorphic in $\operatorname{Re}z>\frac{1}{2},$ then the Riemann hypothesis has a positive solution.