Arithmetical functions and irrationality of Lambert series

Abstract

We use a method of Erdos in order to prove the linear independence over Q of the numbers 1, ∑ n = 1+∞1qn2−1, ∑ n = 1+∞nqn2−1 for every q∈Z, with |q|≥2. The main idea consists in considering the two above series as Lambert series. This allows to expand them as power series of 1/q. The Taylor coefficients of these expansions are arithmetical functions, whose properties allow to apply an elementary irrationality criterion, which yields the result.