Moments of gaps between \(k\)-free numbers
type: Item
instance of: scholarly article
title: Moments of gaps between \(k\)-free numbers
zbMATH Open document ID: 0787.11039
published in: Journal of Number Theory
publication date: 1994-05-15T00:00:00Z
review text: Let \(k \geq 2\) be a fixed integer and let \(S_ n = S_ n(k)\) be the \(n\)-th \(k\)-free number. P. Erdős considered a problem of finding a largest \(\gamma\) such that \[ \sum_{s_ n\leq x}(s_{n+1} - s_ n)^ \gamma \ll x\tag{1} \] (obviously, if the inequality holds for \(\gamma_ 0\), it also holds for any \(\gamma \leq \gamma_ 0\)). Since the number of \(k\)-free integers \(\leq x\) is asymptotic to \(x/\zeta(k)\), (1) is the best possible. Erdős proved that if \(k=2\), then one can take \(\gamma = 2\). C. Hooley proved that for any \(k\) one can take \(\gamma = k+1\). The author of this paper improves Hooley's result by showing that (1) holds for any \(\gamma < 2k - 2 + 4/(k + 1)\).
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 205461
zbMATH Keywords: moments
zbMATH Keywords: gaps between \(k\)-free numbers
zbMATH Keywords: number of \(k\)-free integers
author: Sidney W. Graham
reviewed by: G. A. Kolesnik
MaRDI profile type: Publication
full work available at URL: jnth.1993.1039
OpenAlex ID: W1977975816
DOI: 10.1006/JNTH.1993.1039
description: scientific article; zbMATH DE number 205461