#### DOCUMENTA MATHEMATICA, Vol. 18 (2013), 1243-1274

Kazunori Noguchi

The Zeta Function of a Finite Category

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 \cite{Leib}. 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.

2010 Mathematics Subject Classification: Primary 18G30; Secondary 18D30, 30B10, 30B40.

Keywords and Phrases: zeta function of a finite category, Euler characteristics of categories, coverings of small categories, Dedekind conjecture

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