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

Abstract

We prove that if a function $\theta \left(z\right)=\underset{1}{\overset{\infty }{\int }}\frac{\pi \left(t\right)-Li\left(t\right)}{t^{z+1}}dt,$ which is holomorphic in $\left\{\mathrm{Re}z>1\right\}$ holomorphically extends to some simply connected domain $G\subset \left\{\mathrm{Re}z>\frac{1}{2}\right\}$, 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 $\mathrm{Re}z>\frac{1}{2},$ then the Riemann hypothesis has a positive solution.