On some Euler products. II
type: Item
instance of: scholarly article
title: On some Euler products. II
zbMATH Open document ID: 0577.12009
author: Nobushige Kurokawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1984-01-01T00:00:00Z
review text: The result on Euler products alluded to in the previous note by the author [same volume, 335-338 (1984; Zbl 0559.12009)] is formulated in this note. The Euler products are defined in the following situation. Let be given a set P provided with a ''norm'' in a certain sense (think at the set of primes of an algebraic number field) and a map \(\alpha\) from P into some group G (think of p \(\to\) Frobenius automorphism). Then to any polynomial H(g,T) in T whose coefficients are functions of \(g\in G\), supposed to be virtual characters, and congruent to 1 modulo T is assigned the Euler product with general factor H(\(\alpha\) (p), N(p)\({}^{-s})^{-1}\). Now the author gives a sufficient condition on the datum (P,G,\(\alpha)\) in order to have the following. The Euler product corresponding to H has a meromorphic continuation to \({\mathbb{C}}\) if H is of the form \(H(g,T)=\det (1-M(g)T)\), M(g) a unitary matrix, and otherwise the Euler product has \(Re(s)=0\) as a natural boundary. The condition consists of certain properties of the L-function obtained by taking \(H(g,T)=\det (1-\rho (g)T)\), \(\rho\) an irreducible finite dimensional unitary representation of G. An idea of the proof is given, and two examples.
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 3922792
zbMATH Keywords: Euler datum
zbMATH Keywords: Euler products
zbMATH Keywords: norm
zbMATH Keywords: virtual characters
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: natural boundary
zbMATH Keywords: L-function
zbMATH Keywords: unitary representation
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: pjaa.60.365
OpenAlex ID: W4240150191
cites work: On some Euler products. I
DOI: 10.3792/PJAA.60.365
Recommended article: On some Euler products. I
Recommended article: On The Meromorphy of Euler Products (I)
Recommended article: On the Meromorphy of Euler Products (II)
Recommended article: Q3708091
Recommended article: On certain Euler products
Recommended article: On certain Dirichlet series associated with automorphic forms on SL(2,\({\mathbb{C}})\)
Recommended article: Differential operators and the singular values of Eisenstein series
Recommended article: A new way to get Euler products.
Recommended article: Q3052982
Recommended article: Zeta functions and Eisenstein series on classical groups
description: scientific article; zbMATH DE number 3922792