On the $p^λ$ problem
Abstract
Description
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's method and on using new mean value estimates for products of shifted and ordinary (unshifted) Dirichlet polynomials.
35 pages
35 pages