Fractional parts of non-integer powers of primes. II

We continue to study the distribution of prime numbers $p$, satisfying the condition $\{p^{\alpha }\}\in I\subset [0;1)$, in arithmetic progressions. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for $0 < \alpha < 1/9$withthelevelofdistribution$\theta = 2/5 - (3/5) \alpha$, which improves the previous result corresponding to $\theta \le 1\mathrm{/}3$.