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.