A connection between the poles of the zeta function of a recurrence sequence and the module of relations of its roots

Álvaro Serrano Holgado

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arXiv · 2311.08632

A connection between the poles of the zeta function of a recurrence sequence and the module of relations of its roots

Álvaro Serrano Holgado

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Abstract

Answering a question left open in previous research, we study the enumeration of poles of the zeta function $φ(s)$ associated to an integer linear recurrence sequence $\{a_n\}$. This enumeration can count poles more than once, and we prove that this happens if and only if the module of relations of the roots of the recurrence is nontrivial. A review of the existing literature on the module of relations yields a series of sufficient conditions for the enumeration of poles of $φ(s)$ to be injective. All of this is illustrated by examples of both cases.

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BibTeX RIS

Álvaro Serrano Holgado. 2023-11-15. A connection between the poles of the zeta function of a recurrence sequence and the module of relations of its roots. https://arxiv.org/abs/2311.08632

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