On the $p^λ$ problem

Abstract

We deal with the distribution of the fractional parts of $p^λ$, $p$ running over the prime numbers and $λ$ being a fixed real number lying in the interval $(0,1)$. Roughly speaking, we study the following question: Given a real $θ$, how small may $δ>0$ be choosen if we suppose that the number of primes $p\le N$ satisfying ${p^λ-θ<δ}$ is close to the expected one? We improve some results of Balog and Harman on this question for $λ<5/66$ if $θ$ is rational and for $λ<1/5$ if $θ$ is irrational. Our improvement is based on incorporating the zero detection argument into Harman&#39;s method and on using new mean value estimates for products of shifted and ordinary (unshifted) Dirichlet polynomials.