Refinements of Erdős'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 Erdős 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 \[ \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{σ(n)}} \quad\text{and}\quad \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{ϕ(n)}} \] are both irrational, where $d(n)$, $σ(n)$, and $ϕ(n)$ denote the number of divisors, the sum of divisors, and Euler's totient functions, respectively.