Artin L-function

In mathematics, Artin L-functions are a type of Dirichlet series defined for finite extensions of number fields, encoding informations about linear representations of Galois group, ramification of prime ideals and distribution of absolute norms of ideals.

These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. They play important role in modern number theory and are generalizations of better-known functions like Dedekind zeta functions or Dirichlet L-functions. Some of their expected properties turned out to be difficult to prove.

One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm basis.