Study of \(L(s,\chi)/\pi^s\) for \(L\)-functions relative to \(\mathbb{F}_q((T^{-1}))\) and associated to characters of degree 1
type: Item
instance of: scholarly article
title: Study of \(L(s,\chi)/\pi^s\) for \(L\)-functions relative to \(\mathbb{F}_q((T^{-1}))\) and associated to characters of degree 1
zbMATH Open document ID: 0994.11027
author: Gilles Damamme
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2000-04-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/224698
review text: The author investigates transcendence questions related to \(L\)-functions in characteristic \(p\). Let \(k={\mathbb F}_q(T)\) be the rational function field over \({\mathbb F}_q\). \textit{L. Carlitz} [Duke Math. J. 1, 137-168 (1935; Zbl 0012.04904)] defined an analogue of \(\pi\) for the field \(k\). Let \(L(s,\chi)\) denote the Goss \(L\)-function associated to a character \(\chi\) [\textit{D. Goss}, Pac. J. Math. 105, 143-181 (1983; Zbl 0571.14010)]. In the paper under review, the author proves that there exist integer values \(s\) and characters \(\chi\) such that \(L(s,\chi)/\pi^s\) is rational, algebraic or transcendent. The case of \(L\)-functions associated to characters of level 1 is discussed in detail.
Mathematics Subject Classification ID: 11J93
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11M38
zbMATH DE Number: 1426377
zbMATH Keywords: Carlitz module
zbMATH Keywords: \(L\)-function in characteristic \(p\)
zbMATH Keywords: transcendence
reviewed by: Igor Potemine
MaRDI profile type: Publication
OpenAlex ID: W2028548968
cites work: Sur la transcendance de la série formelle Π
cites work: Transcendence of the values of the Carlitz zeta function by Wade's method
cites work: On a new type of \(L\)-function for algebraic curves over finite fields
cites work: Irrationality measures and transcendence in positive characteristic
cites work: Q4317303
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.256
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description: scientific article; zbMATH DE number 1426377