  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.
