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On the convergence of the Dirichlet series of an Artin L-function

✍ Scribed by Florin Nicolae; Michael Pohst

Publisher
John Wiley and Sons
Year
2011
Tongue
English
Weight
86 KB
Volume
284
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$K/\mathbb {Q}$\end{document} be a finite Galois extension with the Galois group G, and let Ο‡ be a character of G with the associated Artin L‐function L(s, Ο‡) defined in β„œ(s) > 1 by the Dirichlet series \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sum _{n=1}^\infty \frac{a_n}{n^s}$\end{document} with abscissa of convergence Οƒ~c~. Assume that L(s, Ο‡) is holomorphic in the whole complex plane. If Ο‡(1) = 1 then Οƒ~c~ = 0, and if Ο‡(1) > 1 then \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sigma _c\le \frac{\chi (1)}{2+\chi (1)}$\end{document}.

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