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

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 $n\le x$ not belonging to a set of size $o(x)$, \[ |ω(s(n)) - \log\log s(n)| < ε\log\log s(n) \] and the same is true for $\Omega (s(n))$.

arXiv comment12 pages