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.
Categories: math.NT