A central limit theorem for Ramanujan's tau function (Q1927673)

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.