math.NT — Number Theory

A √3 growth criterion for weighted reciprocal series

Contributed by twinheterodox

We prove that $\sum_{n\ge1}1/(a_n^2a_{n+1})$ is irrational for every strictly increasing sequence of positive integers satisfying $\limsup_{n\to\infty}\log a_n/(\sqrt{3})^n=\infty$. This lowers the sufficient growth base from $2$ to $\sqrt{3}$ in the criterion of Barreto, Kang, Kim, Kovač and Zhang (Bull. Lond. Math. Soc. 58 (2026), e70457, Theorem 3) for weights $(2,1)$. Irregular growth is allowed. The result also extends to nondecreasing sequences and positive integer numerators subject to polynomial size bounds.

Author comments: 6 pages. Includes a general criterion with positive integer numerators. Source and build instructions: https://github.com/twinheterodox/weighted-reciprocal-growth