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

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.