Fractional parts of non-integer powers of primes. II

TL;DR

This paper extends the understanding of the distribution of primes with fractional parts of their non-integer powers, proving an improved Bombieri-Vinogradov type theorem for certain exponents.

Contribution

It establishes a new Bombieri-Vinogradov type result for primes with fractional powers, improving the level of distribution for 0 < α < 1/9.

Findings

Proves an analogue of Bombieri-Vinogradov theorem for α < 1/9

Achieves a higher level of distribution θ = 2/5 - (3/5) α

Improves previous distribution bounds for primes in this context

Abstract

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 $0 < \alpha < 1/9$ with the level of distribution $\theta = 2/5 - (3/5) \alpha$, which improves the previous result corresponding to $\theta \leq 1/3$.