One functional property of the \(\varsigma\)-function of Riemann

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.