Sums of proper divisors follow the Erd\H{o}s--Kac law

Abstract

Let $s(n)=\sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. The second-named author proved that $\omega(s(n))$ has normal order $\log\log{n}$, the analogue for $s$-values of a classical result of Hardy and Ramanujan. We establish the corresponding Erd\H{o}s--Kac theorem: $\omega(s(n))$ is asymptotically normally distributed with mean and variance $\log\log{n}$. The same method applies with $s(n)$ replaced by any of several other unconventional arithmetic functions, such as $\beta(n):=\sum_{p\mid n} p$, $n-\phi(n)$, and $n+\tau(n)$ ($\tau$ being the divisor function).