Linear Independence of Values of a Certain Lambert Series

Let K be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{Q}}$$ \end{document} or an imaginary quadratic number field, and q \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\in$$ \end{document} K an integer with |q| > 1. We give a quantitative version of the linear independence over K of the three numbers 1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\nolimits_{k \geq 1} {1/(q^{2k - 1} + 1),}\, \sum\nolimits_{k \geq 1} {1/(q^{2k - 1} - 1)}$$ \end{document}, and an equivalent power series version. We also mention several open problems.