The irrationality of a number theoretical series

Abstract

Denote by ${\sigma }_k(n)$ the sum of the $k$-th powers of the divisors of $n$, and let $S_k={\sum }_{n\ge 1}\frac{{\sigma }_k(n)}{n!}$. We prove that Schinzel's conjecture H implies that $S_k$ is irrational, and give an unconditional proof for the case $k=3$.