The irrationality of a divisor function series of Erdős and Kac

Abstract

For positive integers $k$ and $n$ let $σ_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erdős and Kac asked whether, for every $k$, the number $α_k = \sum_{n\geq 1} \frac{σ_k(n)}{n!}$ is irrational. It is known unconditionally that $α_k$ is irrational if $k\leq 3$. We prove $α_4$ is irrational.