The irrationality of a divisor function series of Erdős and Kac
Volume 211 / 2023
Acta Arithmetica 211 (2023), 193-228
MSC: Primary 11J72.
DOI: 10.4064/aa220927-1-9
Published online: 8 November 2023
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 unconditionally that $\alpha _k$ is irrational if $k\leq 3$. We prove that $\alpha _4$ is irrational.
