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4 - L-functions from number theory

Published online by Cambridge University Press:  28 April 2020

Bruno Kahn
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche
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Summary

This chapter returns to more elementary mathematics, introducing Dirichlet, Hecke, and Artin L-functions. A proof of Dirichlet’s theorem on arithmetic progressions is given, by the method expounded by Serre; it would however be a shame to omit Dirichlet’s original method, which gave additional information and anticipated the analytic class number formulae. The two main generalisations of Dirichlet’s L-functions are then introduced: those of Hecke and Artin. Hecke’s main theorem is stated without proof: existence of an analytic continuation and a functional equation, and it is then explained how Artin and Brauer derived the same results for non-abelian L-functions.

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Publisher: Cambridge University Press
Print publication year: 2020

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