On the arithmetic properties of complex values of Hecke-Mahler series. I. The rank one case

Here we characterise, in a complete and explicit way, the relations of algebraic dependence over $\mathbb{Q}$ of complex values of Hecke-Mahler series taken at algebraic points ${\underline{u}}_1,...,{\underline{u}}_m$ of the multiplicative group ${𝔾}_{\mathrm{m}}^2\left(ℂ\right)$, under a technical hypothesis that a certain sub-module of ${𝔾}_{\mathrm{m}}^2\left(ℂ\right)$ generated by the ${\underline{u}}_i$’s has rank one (rank one hypothesis). This is the first part of a work, announced in [Pel1], whose main objective is completely to solve a general problem on the algebraic independence of values of these series.