Linear Independence of Values of a Certain Lambert Series

Abstract

Let K be $${\mathbb{Q}}$$ or an imaginary quadratic number field, and q $$\in$$ K an integer with |q| > 1. We give a quantitative version of the linear independence over K of the three numbers 1, $$\sum\nolimits_{k \geq 1} {1/(q^{2k - 1} + 1),}\, \sum\nolimits_{k \geq 1} {1/(q^{2k - 1} - 1)}$$ , and an equivalent power series version. We also mention several open problems.