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.