Algebraic independence results on the generating Lambert series of the powers of a fixed integer

In this paper, the algebraic independence of values of the functionGd(z) := ∑ h≥0 z dh/(1− zdh ), d>1 a fixed integer, at non-zero algebraic points in the unit disk is studied. Whereas the case of multiplicatively independent points has been resolved some time ago, a particularly interesting case of multiplicatively dependent points is considered here, and similar results are obtained for more general functions. The main tool is Mahler’s method reducing the investigation of the algebraic independence of numbers (over Q) to the one of functions (over the rational function field) if these satisfy certain types of functional equations.