### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Documenta Mathematica*,

*18*(2013), 1243-1274.

**The zeta function of a finite category.** / Noguchi, Kazunori.

Research output: Contribution to journal › Article

*Documenta Mathematica*, vol. 18, no. 2013, pp. 1243-1274.

}

TY - JOUR

T1 - The zeta function of a finite category

AU - Noguchi, Kazunori

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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.

AB - 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.

KW - Coverings of small categories

KW - Dedekind conjecture

KW - Euler characteristics of categories

KW - Zeta function of a finite category

UR - http://www.scopus.com/inward/record.url?scp=84918806386&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84918806386&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84918806386

VL - 18

SP - 1243

EP - 1274

JO - Documenta Mathematica

JF - Documenta Mathematica

SN - 1431-0635

IS - 2013

ER -