Carlitz \(\zeta\)-function and automata
type: Item
instance of: scholarly article
title: Carlitz \(\zeta\)-function and automata
zbMATH Open document ID: 0784.11025
author: Valérie Berthé
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1994-03-09T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: The author gives a new proof of the transcendence of the values \(\zeta(s)/\Pi^ s\) for integers \(1\leq s\leq q-2\), where \(\zeta(s)\) is the Carlitz zeta function for the rational function field over \(\mathbb{F}_ q\) and \(\Pi\) is the analogue for the Carlitz module of \(\pi\). The transcendece of these zeta values is a special case of Yu's general theorem, which gives the transcendence of \(\zeta(s)/\Pi^ s\) for all integers \(s\) which are not divisible by \(q-1\) [\textit{J. Yu}, Ann. Math. 134, 1-23 (1991; Zbl 0734.11040)]. The proof in the paper is based on a theorem of \textit{G. Christol}, \textit{T. Kamae}, \textit{M. Mendès-France} and \textit{G. Rauzy} [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)] which gives a criterion for a formal series \(\sum a_ n x^{-n}\in \mathbb{F}_ q((1/x))\) to be algebraic over \(\mathbb{F}_ q(x)\).
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 68Q45
zbMATH DE Number: 484161
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: transcendence
zbMATH Keywords: Carlitz zeta function
zbMATH Keywords: Carlitz module
zbMATH Keywords: zeta values
reviewed by: Jeremy Teitelbaum
MaRDI profile type: Publication
OpenAlex ID: W2315711533
cites work: Sur la transcendance de la série formelle Π
cites work: Suites algébriques, automates et substitutions
cites work: Q3772304
cites work: Q3807306
cites work: Transcendence of the values of the Carlitz zeta function by Wade's method
cites work: Certain quantities transcendental over \(\text{GF}(p^n,x)\)
cites work: Transcendence and special zeta values in characteristic p
DOI: 10.5802/JTNB.78
description: scientific article; zbMATH DE number 484161