  Let $\alpha > 0$ be any fixed non-integer, $I$ be any subinterval of $[0;
1)$. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for the
set of primes $p$ satisfying the condition $\{ p^{\alpha} \} \in I$. This
strengthens the previous result of Gritsenko and Zinchenko.
