The goal of this article is to associate a $p$-adic analytic function to the Euler constants ${\gamma }_p(a,F)$, study the properties of these functions in the neighborhood of $s=1$ and introduce a $p$-adic analogue of the infinite sum ${\sum }_{n\ge 1}f\left(n\right)/n$ for an algebraic valued, periodic function $f$. After this, we prove the theorem of Baker, Birch and Wirsing in this setup and discuss irrationality results associated to $p$-adic Euler constants generalising the earlier known results in this direction. Finally, we define and prove certain...

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.