The zeta function of a finite category

TL;DR

This paper introduces a new zeta function for finite categories, explores its properties, and proposes a conjecture linking it to the Euler characteristic, verified in specific cases.

Contribution

It defines the zeta function for finite categories and formulates a conjecture relating it to the Euler characteristic, with verification in special cases.

Findings

Conjecture holds for finite groupoids and acyclic categories.

Verification for categories with 2 objects.

Confirmed for categories satisfying certain conditions.

Abstract

We define the zeta function of a finite category. And we propose a conjecture which states the relationship between the Euler characteristic of finite categories and the zeta function of finite categories. This conjecture is verified when categories are finite groupoids, finite acyclic categories, categories with 2-objects and finite categories satisfying certain condition.