Generalized Dirichlet series of n variables associated with automatic sequences

TL;DR

This paper establishes conditions for the meromorphic continuation of multivariable Dirichlet series linked to automatic sequences, exploring their holomorphic properties and related infinite product equivalences.

Contribution

It provides a necessary and sufficient condition for the meromorphic continuation of Dirichlet series with automatic sequence coefficients and polynomial denominators.

Findings

Characterization of meromorphic continuation conditions

Examples of holomorphic continuation for specific cases

Connections between Dirichlet series and infinite products

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.