On the distribution of numbers of the form \(\sigma(n)/n\) and on some related questions
type: Item
instance of: scholarly article
title: On the distribution of numbers of the form \(\sigma(n)/n\) and on some related questions
zbMATH Open document ID: 0291.10040
published in: Pacific Journal of Mathematics
publication date: 1974-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pjm/1102912212
review text: An arithmetic function \(f\) is said to have a distribution function, if for any \(c\) the density \(g(c)\) of integers satisfying \(f(n)<c\) exists and \(g(- \infty)=0\), \(g(\infty)=1\). Let \(f(n)= \sigma(n)/n\), where \(\sigma\) is the sum of divisors function. Then the distribution function \(g\) is known to exist, is continuous and monotonic but purely singular. Let \(F(x;a,b)\) be the number of integers \(n\leq x\) satisfying \(a\leq \sigma(n)/n<b\). The author proves the theorem: There is an absolute constant \(c_1\) so that for \(x>t\) \(F \left(x;a,a+{1 \over t} \right) <c_1x/\ln t\), where apart from the constant \(c_1\) the inequality is best possible. Further from the author's work can be derived some best possible estimates for \(g \left(c+{1 \over t} \right) -g(t)\) for the case of \(\sigma(n)/n\). The author also refers to the relevant problems of abundant numbers and of amicable pairs of numbers. Further he deals with the case where \(\sigma\) is replaced by Euler's \(\varphi\)-function and sharpens some earlier known results.
Mathematics Subject Classification ID: 11N60
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3455353
zbMATH Keywords: sum of divisors function
zbMATH Keywords: distribution function
zbMATH Keywords: Euler's totient function
zbMATH Keywords: abundant numbers
zbMATH Keywords: amicable pairs of numbers
MaRDI profile type: Publication
OpenAlex ID: W2167482555
cites work: Q5181769
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DOI: 10.2140/PJM.1974.52.59
author: P. Erdős
description: scientific article; zbMATH DE number 3455353