The value distribution of Artin \(L\)-series and zeros of zeta-functions.
type: Item
instance of: scholarly article
title: The value distribution of Artin \(L\)-series and zeros of zeta-functions.
zbMATH Open document ID: 1040.11080
DOI: 10.1016/S0022-314X(02)00048-3
author: Hartmut Bauer
published in: Journal of Number Theory
publication date: 2003-04-28T00:00:00Z
review text: Let \(K/\mathbb{Q}\) be a finite, Galois extension of \(\mathbb{Q}\), with Galois group \(G= \text{Gal}(K/\mathbb{Q})\). The main purpose of this paper is to obtain a generalization of a theorem of Voronin, and others (see below and the references in the paper) on the distribution of the values of the Riemann zeta-function and Dirichlet \(L\)-series to the case of the Artin \(L\)-functions \(L(s,\chi_j,K/\mathbb{Q})\) associated with a set of linearly independent characters \(\chi_1,\dots, \chi_n\) of \(G\). More precisely, the author proves the following theorem, which is the principal result of this paper and whose many applications are explained and developed in it. With the foregoing notation, denote by \(k\) the order of \(G\) and let \(f_1(s),\dots, f_n(s)\) be holomorphic functions on the set of \(s\in\mathbb{C}\) such that \(| s|< r\), which is continuous on the disc \(| s|\leq r\), where \(r\) is some fixed number such that \(0< r<{1\over 4k}\). Suppose that \(f_j(s)\neq 0\) on \(| s|\leq r\). Then the author's main theorem (Theorem 1.1 of the paper) states that: for every \(\varepsilon> 0\), there is a set \(A_\varepsilon\subset \mathbb{R}\) such that \[ \liminf_{T\to\infty}\, {\text{vol} (A_\varepsilon \cap (0,T))\over T}> 0, \] and for \(j= 1,2,\dots, n\) and any \(t\in A_\varepsilon\) and \(| s|\leq r\), \[ \Biggl| L\Biggl(s+ 1-{1\over 4k}+ it,\,\chi_j, K/\mathbb{Q} \Biggr)\Biggr|< \varepsilon, \] where \(L(z,\chi_j, K/\mathbb{Q})\) denotes the Artin \(L\)-functions. Later in the paper, the author also proves a similar result for a single Artin \(L\)-function, \(L(s,\phi,K/L)\) of a normal extension \(K/L\) of number fields, where, in general, \(\phi\) denotes a non-Abelian character of \(\text{Gal}(K/L)\). That last result may be used to derive, by an appeal to class field theory, a theorem of \textit{H. Mishou} [Acta Arith. 98, 395--410 (2001; Zbl 0982.11050)]. The author also refers to a result of \textit{H. Davenport} and \textit{H. Heilbronn} [On t…
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 1902879
zbMATH Keywords: Artin \(L\)-functions
zbMATH Keywords: universality
zbMATH Keywords: zeta-functions
MaRDI profile type: Publication
cites work: Q5531759
cites work: Q5535091
cites work: Q3955331
cites work: The distribution of values of Artin \(L\)-functions and applications
cites work: On the Distribution of the Values of the Riemann Zeta Function
cites work: Beziehungen zwischen Klassenzahlen von Teilkörpern eines galoisschen Körpers. Herrn Professor Dr. Erhard Schmidt in dankbarer Verehrung zum 75. Geburtstag
cites work: Q5527114
cites work: The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
cites work: The universality theorem for \(L\)-functions associated with ideal class characters
cites work: Q3997223
cites work: Q4001587
cites work: Q4760427
cites work: Q4126536
cites work: Q5803461
description: scientific article; zbMATH DE number 1902879