On the proximity of divisors of integers
type: Item
instance of: scholarly article
title: On the proximity of divisors of integers
zbMATH Open document ID: 0935.11032
author: André Stef
author: Abdelaziz Raouj
published in: Journal of Number Theory
publication date: 1999-11-30T00:00:00Z
review text: The authors consider the interesting problem of finding conditions on \(\lambda\geq 0\) such that, for almost all \(r\)-tuples \((n_1,\dots, n_r)\) of positive integers, there exist divisors \(d_j|n_j\) \((1\leq r)\) with \[ 0<\log (d_j/d_1)\leq (\log n_1)^{-\lambda}, \qquad 2\leq j\leq r. \] \textit{A. Raouj} considered the case \(r=2\) in [Acta Arith. 69, 121-152, 171-188 (1995; Zbl 0821.11045)] where it is shown that the statement above holds for \(\lambda< \lambda_2^*= \log 4-1\) and fails for \(\lambda> \lambda_2^*\). In this paper the authors tackle the cases \(r=3\), \(r\geq 4\). They prove that when \(r=3\) the critical value of \(\lambda\) is \(\lambda_3^*= -1+(3/2)\log 2\), in the sense that the statement above holds for \(\lambda< \lambda_3^*\) and fails for \(\lambda> \lambda_3^*\). However when \(r\geq 4\) they show that the statement fails for all \(\lambda\geq 0\). In all cases an estimate of the number of \(r\)-tuples \((n_1,\dots, n_r)\) with \(n_j\leq x_j\) that lie in the exceptional set is obtained. Inevitably the proof is very technical and depends on a number of subsidiary results.
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1310085
zbMATH Keywords: divisors of integers of an \(r\)-tuple
zbMATH Keywords: nearness of divisors
reviewed by: Eira J. Scourfield
MaRDI profile type: Publication
full work available at URL: jnth.1998.2355
OpenAlex ID: W2049254787
cites work: On the density of certain sequences of integers
cites work: Q4716269
cites work: Q5515488
cites work: On a result of R. R. Hall
cites work: Q4040373
cites work: On the set of divisors of an integer
cites work: Sur la densité de certains ensembles de multiples, 1
cites work: A Brun-Titschmarsh theorem for multiplicative functions.
cites work: Q3123961
cites work: Q4332879
DOI: 10.1006/JNTH.1998.2355
Recommended article: The divisor problem for arithmetic progressions
Recommended article: Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2
Recommended article: THE DISTRIBUTION OF INTEGERS WITH AT LEAST TWO DIVISORS IN A SHORT INTERVAL
Recommended article: Linear forms in the logarithms of three positive rational numbers
Recommended article: Q4182569
Recommended article: Q4249696
Recommended article: Q3820699
Recommended article: Composite positive integers with an average prime factor
Recommended article: On the difference of the number of prime divisors of consecutive numbers
Recommended article: Sets of multiples of finite sequences
description: scientific article; zbMATH DE number 1310085