Refinements of Erdős's irrationality criterion for certain sparse infinite series
Hajime Kaneko, Yuta Suzuki, Yohei Tachiya
TL;DR
The paper develops a Pisot/Salem-number–based refinement of Erdős's irrationality criterion for sparse infinite series and applies it to a broad class of arithmetic-function–driven sums. It proves a fundamental lemma showing that, under precise tail-decay and interlacing conditions for algebraic-integer coefficients in $\mathbb{Q}(q)$, certain power-series sums cannot be rational or algebraic over $\mathbb{Q}(q)$. The authors then derive a prepared version of the criterion and use it to obtain concrete irrationality results, including generalized Erdős-type results, Minkowski-sum irrationalities, degree-ell irrationalities, and Liouville-type transcendence statements for sums involving Euler’s totient function, the divisor function, and their powers. These results yield explicit irrationality (and in some cases transcendence) conclusions for sums such as $\sum_{n\ge1} \frac{d(n)^k}{t^{\sigma(n)}}$ and $\sum_{n\ge1} \frac{d(n)^k}{t^{\varphi(n)}}$, as well as structured corollaries for sums over $\sigma$ and $\varphi$ images, with implications for nonzero-digit density in algebraic representations. The work significantly broadens the toolkit for irrationality and transcendence of sparse series in arithmetic settings, linking Erdős-type methods with Pisot/Salem dynamics and classical divisor–totient functions.
Abstract
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 $t\ge2$ and $k\geq0$, the numbers \[ \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{σ(n)}} \quad\text{and}\quad \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{φ(n)}} \] are both irrational, where $d(n)$, $σ(n)$, and $φ(n)$ denote the number of divisors, the sum of divisors, and Euler's totient functions, respectively.