On the arithmetic properties of complex values of Hecke-Mahler series

Here we characterise in a complete and explicit way the relations of algebraic dependence over $\text{QQ}$ of complex values of Hecke-Mahler series taken at algebraic points of the multiplicative group ${\text{GG}}_m^2(\text{CC})$. Our result contains previous theorems by Loxton and van der Poorten, Mahler, and Masser.