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 and n let \sigma _k(n) denote the sum of the kth 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.
