Linear combinations of \(\zeta(s)/ \Pi^ s\) over \(F_ q(x)\) for \(1\leq s\leq q-2\)
type: Item
instance of: scholarly article
title: Linear combinations of \(\zeta(s)/ \Pi^ s\) over \(F_ q(x)\) for \(1\leq s\leq q-2\)
zbMATH Open document ID: 0853.11062
published in: Journal of Number Theory
publication date: 1997-01-09T00:00:00Z
review text: Let \(k= F_q (x)\) be the rational function field over the finite field with \(q\) elements. The author proves that \(k\)-linear combinations of the values \(\zeta (s)/ \Pi^s\), for \(1\leq s\leq q-2\) and \(q\neq 2\), are transcendental over \(k\). Here \(\zeta (s)\) is the Carlitz zeta function for \(k\) and \(\Pi\) is the period of the Carlitz module. The proof in the paper is an application of a theorem of \textit{G. Christol}, \textit{T. Kamae}, \textit{M. Méndes-France}, and \textit{G. Rauzy} [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)] which gives a criterion derived from the theory of automata for a formal series \(\sum a_n x^{-n}\in F_q ((1/x))\) to be algebraic over \(F_q (x)\). See also the author's earlier paper [J. Théor. Nombres Bordx. 5, No. 1, 53-77 (1993; Zbl 0784.11025)] which considers the transcendence of the individual values \(\zeta (s)/ \Pi^s\) from a similar point of view, and work of \textit{J. Yu} [Ann. Math., II. Ser. 134, 1-23 (1991; Zbl 0734.11040)] which gives general results on transcendence of special values in the function field setting.
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 798702
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: Carlitz zeta function
zbMATH Keywords: Carlitz module
zbMATH Keywords: transcendence of special values
zbMATH Keywords: function field
author: Valérie Berthé
reviewed by: Jeremy Teitelbaum
MaRDI profile type: Publication
full work available at URL: jnth.1995.1091
OpenAlex ID: W2040417075
DOI: 10.1006/JNTH.1995.1091
description: scientific article; zbMATH DE number 798702