The irrationality of a number theoretical series

TL;DR

This paper investigates the irrationality of a specific number series involving divisor sums, proving its irrationality under certain conjectures and unconditionally for a special case.

Contribution

It links Schinzel's conjecture H to the irrationality of the series and provides an unconditional proof for the case when k=3.

Findings

Proves that Schinzel's conjecture H implies the series is irrational.

Provides an unconditional proof of irrationality for the case k=3.

Establishes a connection between number theory conjectures and series irrationality.

Abstract

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