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arXiv · 2209.11124

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

Kyle Pratt
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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.

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Kyle Pratt. 2022-09-22. The irrationality of a divisor function series of Erd\H{o}s and Kac. https://arxiv.org/abs/2209.11124

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