A central limit theorem for partitions involving generalised divisor functions

TL;DR

This paper proves a central limit theorem for the number of parts in restricted partitions involving the generalized divisor function, extending previous work to a broader class of functions using analytic number theory techniques.

Contribution

It establishes a CLT for partitions with parts weighted by the generalized divisor function, generalizing prior results for power functions and employing Dirichlet series analysis.

Findings

Proves a CLT for the number of summands in partitions involving (n)=(n) for divisor functions.

Analyzes the Dirichlet series (n+1)/n^s for (n)=(n) using Ramanujan sums.

Employs Euler product analysis to support the CLT proof.

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.