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 $S_s(n):={\sum }_{d\mid n}s(d)$. 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 ${\mu }_k$ denotes the $k^{th}$ moment of $S_s(n)/n$, we show that $${\mu }_k^{1/k}\sim e^{2\gamma }(\log k{)}^2$$ as $k\to \infty$, where $\gamma$ is the Euler-Mascheroni constant.

A typo in the definition of S_s(n) has been fixed. We now give an asymptotic for mu_k (previously, we just gave an upper bound). The introduction has been rewritten. 16 pages