The limiting distribution of the divisor function
type: Item
instance of: scholarly article
title: The limiting distribution of the divisor function
zbMATH Open document ID: 1233.11085
published in: Journal of Number Theory
publication date: 2009-05-28T00:00:00Z
review text: This article is concerned with the behaviour of three very closely related multiplicative functions, namely the generalised divisor function \[ \sigma_{\alpha}(n) := \sum_{d \mid n} d^{\alpha}, \] Jordan's totient function \(J_{\alpha}(n) := n^{\alpha}\prod_{p \mid n}(1-p^{-\alpha})\), and a generalised Dedekind psi function, \(\Psi_{\alpha}(n) := n^{\alpha} \prod_{p \mid n}(1+p^{-\alpha})\). Here \(\alpha\) is a real parameter. The author studies limiting frequencies such as \[ A_{\alpha}(t) := \lim_{N \to \infty} \frac{1}{N} \#\left\{n \leq N : \frac{\sigma_{\alpha}(n)}{n^{\alpha}} \geq t\right\}, \] aiming to establish their behaviour as functions of \(t\). The main result proved is the following: If \(\varepsilon > 0\) is fixed then \[ A_{\alpha}(t) = \exp\left\{ -\left(\text{Li}^{-1}(\log t)\right)^{1/(1-\alpha)}\left(1+O_{\varepsilon}\left(\frac{(1-\alpha)^{2}}{\alpha(\log\log t)^{2}}\right)\right)\right\}, \] uniformly for \[ t \geq \exp\{e^{e}\}, \qquad \frac{\log\log\log t + \log\log\log\log t}{\log\log t} \leq \alpha \leq 1 - \frac{1}{(\log t)^{1/3-\varepsilon}}. \] The same is true for the limiting frequencies of \(n^{\alpha}/J_{\alpha}(n)\) and \(\Psi_{\alpha}(n)/n^{\alpha}\). Here \(\text{Li}^{-1}\) denotes the inverse function of the logarithmic integral \(\text{Li}(x) := \int_2^x \frac{du}{\log u}\). The proof of this theorem uses the methods of the author's previous article [Proc. Am. Math. Soc. 135, No. 9, 2677--2681 (2007; Zbl 1169.11042)], in which \(A_{1}(t)\) was studied. More specifically, the author bounds the limiting frequencies from below by noting that any number divisible by the product of all the primes less than \(y\), for suitable \(y=y(\alpha,t)\), will satisfy \(\sigma_{\alpha}(n)/n^{\alpha} \geq t\). The resulting lower bound \(1/\prod_{p \leq y} p = e^{-y(1+o(1))}\) is shown to be essentially sharp by a moment method, or (in more number-theoretic language) by using something like Rankin's trick. To be more explicit, the author note…
reviewed by: Adam J. Harper
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N60
zbMATH DE Number: 5559089
zbMATH Keywords: natural density
zbMATH Keywords: limiting distribution
zbMATH Keywords: divisor function
zbMATH Keywords: Jordan's totient function
zbMATH Keywords: Dedekind's psi function
author: Andreas Weingartner
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.01.025
OpenAlex ID: W1975063662
cites work: Some remarks about additive and multiplicative functions
cites work: Q4272975
cites work: The distribution functions of 𝜎(𝑛)/𝑛 and 𝑛/𝜑(𝑛)
DOI: 10.1016/J.JNT.2009.01.025
description: scientific article; zbMATH DE number 5559089