A statistical investigation of a divisor-sum function

TL;DR

This paper investigates the statistical properties of a divisor-sum function related to practical numbers, proving the existence of a continuous distribution and density of its normalized values, and introducing a novel analytical method.

Contribution

It introduces a new method for analyzing divisor-sum functions, demonstrating their distributional properties and density, which advances understanding of their statistical behavior.

Findings

$S_s(n)/n$ has a continuous asymptotic distribution function.

Values of $S_s(n)/n$ are dense in $[0, \, \infty)$.

Established mean values and bounds for higher moments of $S_s(n)/n$.

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. 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 also establish mean value computations for $S_s(n)$ and $S_s(n)/n$, and provide uniform bounds for the higher order moments of $S_s(n)/n$. The main novelty in this paper is that we highlight a new method of Lebowitz-Lockard and Pollack that is useful for showing that certain functions have a continuous distribution function where classical methods sometimes fail.