The irrationality of a divisor function series of Erd\H{o}s and Kac

TL;DR

This paper proves the irrationality of the series sum involving the sum of fourth powers of divisors divided by factorials, extending known results for lower powers.

Contribution

It establishes the irrationality of the series for the case k=4, which was previously unresolved.

Findings

Proves  is irrational

Extends irrationality results from k to k

Supports conjecture for all k

Abstract

For positive integers $k$ and $n$ let $\sigma_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erd\H{o}s 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 $\alpha_4$ is irrational.