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

We prove that if a function $\theta \left(z\right)={\int }_1^{\mathrm{\infty }}\frac{\pi \left(t\right)-Li\left(t\right)}{t^{z+1}}dt,$ which is holomorphic in $\{\mathrm{Re}z>1\}$ holomorphically extends to some simply connected domain $G\subset \{\mathrm{Re}z>\frac{1}{2}\}$, then the $\varsigma \left(z\right)-$function of Riemann has no zeros in this domain, $\varsigma \left(z\right)\mathrm{\ne }0\mathrm{\forall }z\in G\mathrm{.}$ 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.