The zeta function of a recurrence sequence of arbitrary degree

TL;DR

This paper studies the zeta function associated with sequences satisfying linear recurrence relations of any degree, establishing its meromorphic continuation, pole structure, and rational values at negative integers, with applications to known examples.

Contribution

It provides a general framework for the meromorphic continuation and explicit pole and residue formulas for the zeta function of arbitrary recurrence sequences.

Findings

Meromorphic continuation of the zeta function established

Explicit formulas for poles and residues derived

Finite values at negative integers shown to be rational

Abstract

We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex plane, giving explicit formulas for its pole set and residues, as well as for its finite values at negative integers, which are shown to be rational numbers. To illustrate the results, we focus on some concrete examples which have also been studied previously by other authors.