The limiting distribution of the divisor function
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 notes (roughly speaking) that \[ A_{1}(t) \leq \lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N} \left(\frac{\sigma_{\alpha}(n)}{tn^{\alpha}}\right)^{s} \] for any \(s \geq 0\), and using a result about the mean values of multiplicative functions, and choosing \(s\) suitably, finds the right hand side is \(\leq e^{-y(1+o(1))}\). The reviewer found this article to be very clearly written and easy to read. Although there are quite a lot of calculations in the article (mostly aimed at finding an explicit estimation of \(y=y(\alpha,t)\)), these are handled meticulously and the reviewer only spotted two very minor errors. Strictly speaking, Lemma 1 in the article is only valid when \(x^{1-\alpha} \geq 2\), since otherwise \(\text{Li}(x^{1-\alpha}) = 0\). Moreover, in line (3) of the proof of Lemma 1 the author presumably means to say that \(R(x)=\exp\{\sqrt{\log x}\}\). The expression that the author obtains for the limiting distribution \(A_{\alpha}(t)\) looks like a kind of (Type II) extreme value distribution, and the expression that he quotes for \(A_{1}(t)\) looks like a kind of (Type I) extreme value distribution. In the reviewer's opinion, it would be nice if the results of this article could be shown to fit into a general probabilistic framework producing extreme value distributions (just as the famous Erdős--Kac theorem, which is related to the \(\alpha = 0\) case of the functions studied here, is seen to be a probabilistic number theory analogue of a central limit theorem).
scientific article; zbMATH DE number 5559089