Refinements of Erd\H{o}s's irrationality criterion for certain sparse infinite series

TL;DR

This paper develops new irrationality criteria for sparse power series and applies them to prove the irrationality of certain infinite series involving divisor functions and classical arithmetic functions.

Contribution

It generalizes Erdős's irrationality criterion and establishes new irrationality results for series involving divisor, sum of divisors, and Euler's totient functions.

Findings

Proves the irrationality of series involving divisor functions for integers t ≥ 2.

Extends Erdős's criterion to broader classes of sparse series.

Demonstrates irrationality of series with divisor-related functions for various parameters.

Abstract

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.