Sums of proper divisors follow the Erdős--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 $\ddot{I}‰(s(n))$ has normal order $\log \log n$, the analogue for $s$-values of a classical result of Hardy and Ramanujan. We establish the corresponding Erdős--Kac theorem: $\ddot{I}‰(s(n))$ 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 ${\hat{I}}^2(n):={\sum }_{p\mid n}p$, $n-\ddot{I}•(n)$, and $n+\ddot{I}„(n)$ ($\ddot{I}„$ being the divisor function).

12 pages