On the number of prime factors of values of the sum-of-proper-divisors function

TL;DR

This paper extends classical results on the distribution of prime factors to the sum-of-proper-divisors function, showing that the typical number of prime factors of s(n) behaves similarly to that of n.

Contribution

It proves that the normal order of the number of prime factors of s(n) aligns with log log s(n), generalizing Hardy and Ramanujan's results to a new arithmetic function.

Findings

The number of prime factors of s(n) concentrates around log log s(n).

The result holds for almost all n up to x, excluding a negligible set.

Similar behavior is established for both ω(s(n)) and Ω(s(n)).

Abstract

Let $\omega(n)$ (resp. $\Omega(n)$) denote the number of prime divisors (resp. with multiplicity) of a natural number $n$. In 1917, Hardy and Ramanujan proved that the normal order of $\omega(n)$ is $\log\log n$, and the same is true of $\Omega(n)$; roughly speaking, a typical natural number $n$ has about $\log\log n$ prime factors. We prove a similar result for $\omega(s(n))$, where $s(n)$ denotes the sum of the proper divisors of $n$: For any $\epsilon > 0$ and all $n \leq x$ not belonging to a set of size $o(x)$, \[ |\omega(s(n)) - \log\log s(n)| < \epsilon \log\log s(n) \] and the same is true for $\Omega(s(n))$.