Les ensembles de multiples et la densité divisorielle. (Sets of multiples and divisor density)
type: Item
instance of: scholarly article
title: Les ensembles de multiples et la densité divisorielle. (Sets of multiples and divisor density)
zbMATH Open document ID: 0581.10030
DOI: 10.1016/0022-314X(86)90014-4
author: Gérald Tenenbaum
author: Richard R. Hall
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: The paper is centered around a conjecture of Erdős, which states that there exists a \(\lambda >1\) such that almost every integer has a divisor in one of the intervals \((\exp (k^{\lambda}),\quad 2 \exp (k^{\lambda})],\) \(k\geq 1\). The authors prove this conjecture, and in fact show that it holds with any \(\lambda <\lambda_ 0=1.3145... \). The proof depends on the notions of ''divisor density'' and ''uniform distribution on divisors''. A function \(f: {\mathbb{N}}\to {\mathbb{R}}\) is said to be uniformly distributed on divisors if the associated discrepancy function \[ \Delta (n;f):=\sup_{0\leq u<v\leq 1}| \sum_{d| n,\quad f(d)\in [u,v) mod 1}1-(v-u)\tau (n)| \] satisfies \(\Delta (n;f)=o(\tau (n))\), as \(n\to \infty\) on a set of asymptotic density one. (Note that in the definition on p. 315 of the paper this last condition is missing and should be added). The authors establish this property in a quantitative form for certain classes of functions f, which include the functions \(n^{\alpha}\) and (log n)\({}^{\alpha}\) \((\alpha >0)\). The latter case is of relevance for the proof of Erdős' conjecture.
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3929106
zbMATH Keywords: sets of multiples
zbMATH Keywords: distribution of divisors
zbMATH Keywords: divisor density
zbMATH Keywords: uniform distribution on divisors
zbMATH Keywords: discrepancy function
reviewed by: Adolf Hildebrand
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3929106