Generalized Dirichlet series of n variables associated with automatic sequences

Abstract: This article consists to give a necessary and sufficient condition of the meromorphic continuity of Dirichlet series defined as $\sum_{x\in \mathbf{N}^n} \frac{a_{x}}{P(x)^s}$, Where $a_{x}$ is a $q$-automatic sequence of $n$ parameters and $P: \mathbf{C}^n \to \mathbf{C}$ a polynomial, such that $P$ does not have zeros on $\mathbf{Q}^{n}_{+}$. And some specific cases of $n=1$ will also be studied in this article as examples to show the possibility to have an holomorphic continuity on the whole complex plane. Some equivalences between infinite products are also built as consequences of these results.