# The zeta function of a finite category

### Kazunori Noguchi

Faculty of Science Shinshu University Matsumoto 390-8621 Japan

## Abstract

We define the zeta function of a finite category. We prove a theorem that states a relationship between the zeta function of a finite category and the Euler characteristic of finite categories, called the series Euler characteristic citeLeib. Moreover, it is shown that for a covering of finite categories, $\map{P}E{B}$, the zeta function of $E$ is that of $B$ to the power of the number of sheets in the covering. This is a categorical analogue of the unproved conjecture of Dedekind for algebraic number fields and the Dedekind zeta functions.

## Cite this article

Kazunori Noguchi, The zeta function of a finite category. Doc. Math. 18 (2013), pp. 1243–1274

DOI 10.4171/DM/427