A statistical investigation of a divisor-sum function

Abstract

The sum of proper divisors function $s(n)$ has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) \coloneqq \sum_{d \mid n} s(d)$. This function arises from a generalization of the practical numbers. Although $S_s(n)$ is neither additive nor multiplicative, we prove that $S_s(n)/n$ has a continuous asymptotic distribution function, and that its values are dense in the interval $[0,\infty)$. We evaluate its mean and establish the existence of all higher moments. Moreover, if $μ_k$ denotes the $k^{th}$ moment of $S_s(n)/n$, we show that $$μ_k^{1/k} \sim e^{2γ} (\log k)^2$$ as $k \rightarrow \infty$, where $γ$ is the Euler-Mascheroni constant.