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Zeros and poles of Artin L-series

Published online by Cambridge University Press:  24 October 2008

Richard Foote  and
V. Kumar Murty
Richard Foote
Affiliation:
Department of Mathematics, University of Vermont, Burlington, VT 05405, U.S.A.
V. Kumar Murty
Affiliation:
Department of Mathematics, Concordia University, Montréal, H3G 1M8, Canada
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Extract

Let E/F be a finite normal extension of number fields with Galois group G. For each virtual character χ of G, denote by L(s, χ) = L(s, χ, F) the Artin L-series attached to χ. It is defined for Re (s) > 1 by an Euler product which is absolutely convergent, making it holomorphic in this half plane. Artin's holomorphy conjecture asserts that, if χ is a character, L(s, χ) has a continuation to the entire s-plane, analytic except possibly for-a pole at s = 1 of multiplicity equal to 〈χ, 1〉, where 1 denotes the trivial character. A well-known group-theoretic result of Brauer implies that L(s, χ) has a meromorphic continuation for all s.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 1 , January 1989 , pp. 5 - 11
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

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