The irrationality of a divisor function series of Erdős and Kac

Abstract

For positive integers $k$ and $n$ let $\sigma _k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erdős and Kac asked whether, for every $k$, the number $\alpha _k = \sum _{n\geq 1} \frac {\sigma _k(n)}{n!}$ is irrational. It is known uncond