Refinements of Erd\H{o}s's irrationality criterion for certain sparse infinite series

Abstract

In this paper, we establish new irrationality criteria for certain sparse power series. As applications of these criteria, we generalize a result of Erd\H{o}s and obtain several irrationality results for various infinite series involving the classical arithmetic functions. For example, we prove that for any integers $t\ge2$ and $k\geq0$, the numbers \[ \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{\sigma(n)}} \quad\text{and}\quad \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{\phi(n)}} \] are both irrational, where $d(n)$, $\sigma(n)$, and $\phi(n)$ denote the number of divisors, the sum of divisors, and Euler's totient functions, respectively.