A central limit theorem for partitions involving generalised divisor functions

Abstract

We define an $f$-restricted partition $p_f(n,k)$ of fixed length $k$ given by the bivariate generating series \begin{align*} Q_f(z,u) \coloneqq 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n =\prod_{k=1}^{\infty}(1+uz^k)^{\Delta_f(k)}, \end{align*} where $\Delta_f(n)=f(n+1)-f(n)$. In this article, we establish a central limit theorem for the number of summands in such partitions when $f(n)=\sigma_r(n)$ denotes the generalised divisor function, defined as $\sigma_r(n)=\sum_{d|n}d^r$ for integer $r\geq 2$. This can be considered as a generalisation of the work of Lipnik, Madritsch, and Tichy, who previously studied this problem for $f(n)=\lfloor{n}^{\alpha}\rfloor$ with $0<\alpha<1$. A key element of our proof relies on the analytic behaviour of the Dirichlet series \begin{align*} \sum_{n=1}^{\infty}\frac{\sigma_r(n+1)}{n^s}, \end{align*} for $\mathrm{Re}(s)>1$. We study this problem employing the identity involving the Ramanujan sum. Furthermore, we analyse the Euler product arising from the above Dirichlet series by adopting the argument of Alkan, Ledoan and Zaharescu.