Fractional parts of non-integer powers of primes

TL;DR

This paper proves an analogue of the Bombieri-Vinogradov theorem for primes whose fractional powers fall within a specific interval, extending previous results and deepening understanding of prime distribution related to non-integer powers.

Contribution

It establishes a new distribution result for primes based on fractional parts of their non-integer powers, strengthening prior work by Gritsenko and Zinchenko.

Findings

Proves an analogue of Bombieri-Vinogradov theorem for primes with fractional powers in a given interval

Extends previous results on the distribution of primes related to non-integer powers

Provides new insights into the distribution of primes in fractional power sequences

Abstract

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.