Zeta functions of categories
type: Item
instance of: scholarly article
title: Zeta functions of categories
zbMATH Open document ID: 0880.11063
author: Nobushige Kurokawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-07-23T00:00:00Z
review text: This paper is a trial of the interpretation of zeta functions from the categorical point of view. Let \(C\) be a category with zero object. We say that an object \(X\) is simple if \(\Hom(X,Y)\) consists of monomorphisms and zero-morphisms for every object \(Y\). The norm \(N(X)\) of an object \(X\) is defined as the cardinality of endomorphisms of \(X\). We say that \(X\) is finite if \(N(X)\) is finite. We denote by \(P(C)\) the isomorphism classes of all finite simple objects of \(C\). The norm is defined for each isomorphism class. We define the zeta function \(\zeta(s,C)\) of \(C\) by \(\zeta(s,C): =\prod_{P \in P(C)} (1-N (P)^{-s})^{-1}\). The author gives some examples and how to unify Hasse zeta and Selberg zeta functions from his point of view.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 18A05
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 1022272
zbMATH Keywords: Hasse zeta functions
zbMATH Keywords: Selberg zeta functions
zbMATH Keywords: category
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: pjaa.72.221
OpenAlex ID: W2069405902
cites work: On some Euler products. I
cites work: Q4733924
cites work: Q3137235
cites work: Q4693151
DOI: 10.3792/PJAA.72.221
description: scientific article; zbMATH DE number 1022272