A central limit theorem for partitions involving generalised divisor function

TL;DR

This work proves a central limit theorem for the number of summands in f-restricted partitions when f is the generalised divisor function $f(n)=\sigma_r(n)$. The authors develop a robust analytic framework combining a bivariate generating function, Mellin transforms, and saddle-point methods to control the shifted Dirichlet-series coefficients that arise from Ramanujan sums. They establish a CLT for the random variable $\varpi_{f,n}$ counting summands, with explicit mean and variance formulas and asymptotics, and provide detailed tail bounds. The methodology extends prior results (e.g., Lipnik–Madritsch–Tichy) to a broader class of multiplicative functions, including Ramanujan sums and divisor functions, via a careful Dirichlet-series analysis and Euler-product techniques. The results illuminate the distributional behavior of partitions under generalized arithmetic weights and yield precise probabilistic descriptions essential for number-theoretic combinatorics.

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)^{Δ_f(k)}, \end{align*} where $Δ_f(n)=f(n+1)-f(n)$. In this article, we establish a central limit theorem for the number of summands in such partitions when $f(n)=σ_r(n)$ denotes the generalised divisor function, defined as $σ_r(n)=\sum_{d|n}d^r$ 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 $f(n)=\lfloor{n}^α\rfloor$ with $0<α<1$. 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)}{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.

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Theorem 2.2
• Theorem 2.3: Converse Mapping flajoletetal
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 3.4