A central limit theorem for Ramanujan's tau function
type: Item
instance of: scholarly article
title: A central limit theorem for Ramanujan's tau function
zbMATH Open document ID: 1277.11094
published in: The Ramanujan Journal
publication date: 2013-01-02T00:00:00Z
review text: Let \(\tau\) be the Ramanujan tau function. By definition, \(\tau(n)\in \mathbb{Z}\) and Lehmer (1947) conjectured that \(\tau(n)=0\) cannot happen. By Deligne's work, now it is known that \(|\tau(n)|/n^{11/2}\leq d(n)\) where \(d(n)\) counts the number of divisors of \(n\). In this paper, the author proves the following central limit theorem: For every \(z\in \mathbb{R}\), \[ (\alpha x)^{-1}\sum_{n\leq x\atop |\beta(n)|n^{-11/2} \leq \exp(A(x)+zB(x))} \tau(n)^2n^{-11} {\longrightarrow} \frac1{\sqrt{2\pi}} \int_{-\infty}^z e^{-u^2/2}\,du, \] as \(x\to \infty\), for \(\beta(n)=\tau(n)\) or \(\beta(n)=r_{24}(n)- \frac{16}{691}\sigma_{11}^*(n)\). Here, \(\alpha>0\) is an explicit constant, \[ A(x) =\frac12\sum_{q\leq x,\, \tau(q)\neq 0 \atop q\, \text{prime power}} \tau(q)^2 q^{-12}\log (\tau(q)^2q^{-11}), \] \[ B(x) =\left(\frac14\sum_{q\leq x,\, \tau(q)\neq 0 \atop q\, \text{prime power}} \tau(q)^2 q^{-12}(\log (\tau(q)^2q^{-11}))^2\right)^{1/2} \geq 0, \] \(r_{24}(n)\) counts the number of representations of \(n\) as a sum of \(24\) squares and \(\sigma_r^*(n) = \sum_{d|n}(-1)^d d^r\). For the proof, the author constructs a Kubilius model so that the left-hand side is linked to some probability measures. Then the convergence follows from the results in probability theory when some conditions are fulfilled, which are verified with tools in analytic number theory. Certainly this interesting and robust method applies to more general context, as is remarked by the author.
reviewed by: Yuk-Kam Lau
Mathematics Subject Classification ID: 11N60
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 6120475
zbMATH Keywords: Ramanujan tau function
zbMATH Keywords: modular function
zbMATH Keywords: central limit theorem
author: Peter D. T. A. Elliott
MaRDI profile type: Publication
full work available at URL: s11139-011-9367-1
OpenAlex ID: W2004232811
cites work: A family of Calabi-Yau varieties and potential automorphy. II.
cites work: La conjecture de Weil. I
cites work: Q3867876
cites work: Q3922753
cites work: The Norms of Compositions of Arithmetic Operators
cites work: Q3122699
cites work: Q5829358
cites work: Q5779465
cites work: Odd values of the Ramanujan $\tau$-function
cites work: Q5774823
cites work: Q5777299
cites work: Quelques applications du théorème de densité de Chebotarev
cites work: AN EXAMPLE IN THE THEORY OF THE SPECTRUM OF A FUNCTION
DOI: 10.1007/S11139-011-9367-1
description: scientific article; zbMATH DE number 6120475