### 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 [BL08]. Moreover, it is shown that for a covering of finite categories, 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.

Original language | English |
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Pages (from-to) | 1243-1274 |

Number of pages | 32 |

Journal | Documenta Mathematica |

Volume | 18 |

Issue number | 2013 |

Publication status | Published - 2013 Jan 1 |

Externally published | Yes |

### Keywords

- Coverings of small categories
- Dedekind conjecture
- Euler characteristics of categories
- Zeta function of a finite category

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Noguchi, K. (2013). The zeta function of a finite category.

*Documenta Mathematica*,*18*(2013), 1243-1274.