On the greatest common divisor of a number and its sum of divisors
type: Item
instance of: scholarly article
title: On the greatest common divisor of a number and its sum of divisors
zbMATH Open document ID: 1286.11152
author: Paul Pollack
published in: Michigan Mathematical Journal
publication date: 2011-05-23T00:00:00Z
review text: Let \(\sigma(n)\) be the sum of divisors of \(n\). For real numbers \(x\geq 1\) and \(A\geq 1\), put \[ G(x,A)=\#\{n\leq x: \gcd(n,\sigma(n))>A\}. \] In the paper under review, the author investigates the behavior of \(G(x,A)\) in various ranges of \(A\) versus \(x\). For example, it is shown that if \(\beta>0\) is fixed, \(x>x_0(\beta)\) is sufficiently large and \(A\geq \exp((\log\log x)^{\beta})\), then \(G(x,A)<x/A^c\), where \(c=c(\beta)>0\). On the other hand, if \(\beta(x)\) is some function tending to zero with \(x\) and \(\varepsilon>0\) is fixed, then provided that \(x>x_0(\varepsilon,\beta)\) and \(2\leq A\leq \exp((\log\log x)^{\beta})\), the estimate \(G(A,x)\geq x/A^{\varepsilon}\) holds. It is also shown that the estimate \[ \sum_{n\leq x} \gcd(n,\sigma(m))\leq x^{1+c_1/{\sqrt{\log\log x}}} \] holds with some positive constant \(c_1\) for all \(x\geq 3\). Furthermore, for all fixed \(\varepsilon>0\), the estimate \(G(x,A)\geq x/A^{1+o(1)}\) holds uniformly for \(2\leq A\leq A^{1-\varepsilon}\). In particular, \(G(x,x^{\delta})=x^{1-\delta+o(1)}\) as \(x\to\infty\), whenever \(\delta\in (0,1)\) is fixed. Some of these results were claimed by Erdős who did not supply proofs. The proofs use an assortment of results from the anatomy of integers such as the distribution of smooth numbers and sieve methods.
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 5899031
zbMATH Keywords: sum of divisors
zbMATH Keywords: greatest common divisor
zbMATH Keywords: applications of sieve methods
reviewed by: Florian Luca
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1301586311
OpenAlex ID: W2004338620
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DOI: 10.1307/MMJ/1301586311
description: scientific article; zbMATH DE number 5899031