On the $p^λ$ problem
| dc.creator | Baier, Stephan | |
| dc.date | 2005-12-19 | |
| dc.date.accessioned | 2026-07-07T06:55:31Z | |
| dc.date.available | 2026-07-07T06:55:31Z | |
| dc.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. | |
| dc.description | 35 pages | |
| dc.identifier | https://arxiv.org/abs/math/0512445 | |
| dc.identifier | http://arxiv.org/abs/math/0512445 | |
| dc.identifier | Acta Arithmetica 113 (2004) 77-101 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/106306 | |
| dc.subject | Number Theory | |
| dc.subject | 11N05, 11M26 | |
| dc.title | On the $p^λ$ problem | |
| dc.type | text |