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

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 $t\ge 2$ and $k\ge 0$, the numbers $$\sum _{n=1}^{\mathrm{\infty }}\frac{d(n{)}^k}{t^{\sigma (n)}}\text{and}\sum _{n=1}^{\mathrm{\infty }}\frac{d(n{)}^k}{t^{\phi (n)}}$$ 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.