Refinements of Erdős's irrationality criterion for certain sparse infinite series

In this paper, we establish new irrationality criteria for certain sparse infinite 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 Formula: see text and Formula: see text, the numbers Formula: see text and Formula: see text are both irrational, where Formula: see text, Formula: see text, and Formula: see text denote the number of divisors, the sum-of-divisors function, and Euler’s totient function, respectively.