A central limit theorem for partitions involving generalised divisor functions

Abstract

We let $p_f(n,k)$ count the $f$-restricted coloured partitions of $n$ with $k$ parts, for a positive integer valued function $f$, given by the bivariate generating series \[ Q_f(z,u) := 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n =\prod_{k=1}^{\infty}(1+uz^k)^{Δ_f(k)}, \] where ${\Delta }_f(n)=|f(n+1)-f(n)|$. A part of size $n$ has ${\Delta }_f(n)$ distinguishable colours, and each coloured part is available at most once. 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 $r\ge 2$. This problem is related to the work of Lipnik, Madritsch, and Tichy on restricted partitions into the indexed values $\lfloor m^{\alpha }\rfloor$, 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{|σ_r(n+1)-σ_r(n)|}{n^s}, \end{align*} for $\mathrm{Re}(s)>r+1$. We study this problem employing the identity involving the Ramanujan sum. Furthermore, we analyse the Euler product arising in this analysis by adopting the argument of Alkan, Ledoan and Zaharescu. The absolute values are handled using an eventual parity pattern, and the required continuation is obtained from the corrected character decomposition.