Artin L-function
In mathematics, an Artin "L"-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group "G". These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. 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. Definition. Given formula_1, a representation of formula_2 on a finite-dimensional complex vector space formula_3, where formula_2 is the Galois group of the finite extension formula_5 of number fields, the Artin formula_6-function: formula_7 is defined by an Euler product. For each prime ideal formula_8 in formula_9's ring of integers, there is an Euler factor, which is easiest to define in the case where formula_8 is unramified in formula_11 (true for almost all formula_8). In that case, the Frobenius element formula_13 is defined as  a conjugacy class in formula_2. Therefore, the characteristic polynomial of formula_15 is well-defined. The Euler factor for formula_16 is a slight modification of the characteristic polynomial, equally well-defined, as rational function in "t", evaluated at formula_18, with formula_19 a complex variable in the usual Riemann zeta function notation. (Here "N" is the field norm of an ideal.) When formula_16 is ramified, and "I" is the inertia group which is a subgroup of "G", a similar construction is applied, but to the subspace of "V" fixed (pointwise) by "I". The Artin L-function formula_7 is then the infinite product  over all prime ideals formula_16 of these factors. As Artin reciprocity shows, when "G" is an abelian group these "L"-functions have a second description (as Dirichlet "L"-functions when "K" is the rational number field, and as Hecke "L"-functions in general). Novelty comes in with non-abelian "G" and their representations. One application is to give factorisations of Dedekind zeta-functions, for example in the case of a number field that is Galois over the rational numbers. In accordance with the decomposition of the regular representation into irreducible representations, such a zeta-function splits into a product of Artin "L"-functions, for  each irreducible representation of "G". For example, the simplest case is when "G" is the symmetric group on three letters. Since "G" has an irreducible representation of degree 2, an Artin "L"-function for such a representation occurs, squared, in the factorisation of the Dedekind zeta-function for such a number field, in a product with the Riemann zeta-function (for the trivial representation) and an "L"-function of Dirichlet's type for the signature representation. More precisely for formula_5 a Galois extension of degree "n", the factorization follows from where formula_29 is the multiplicity of the irreducible representation in the regular  representation, "f" is the order of formula_30 and "n" is replaced by "n/e" at the ramified primes. Since characters are an orthonormal basis of the class functions, after showing some analytic properties of the formula_31 we obtain the Chebotarev density theorem as a generalization of Dirichlet's theorem on arithmetic progressions. Functional equation. Artin L-functions satisfy a functional equation. The function formula_31 is related in its values to formula_33, where formula_34 denotes the complex conjugate representation. More precisely "L" is replaced by formula_35, which is "L" multiplied by certain gamma factors, and then there is an equation of  meromorphic functions with a certain complex number "W"(ρ) of absolute value 1. It is the Artin root number. It has been studied deeply with respect to two types of properties. Firstly Robert Langlands and Pierre Deligne established a factorisation into Langlands–Deligne local constants; this is significant in relation to conjectural relationships to automorphic representations. Also the case of ρ and ρ* being equivalent representations is exactly the one in which the functional equation has the same L-function on each side. It is, algebraically speaking, the case when ρ is a real representation or quaternionic representation. The Artin root  number is, then, either +1 or −1. The question of which sign occurs is linked to Galois module theory . The Artin conjecture. The Artin conjecture on Artin L-functions states that the Artin L-function formula_31 of a non-trivial irreducible representation ρ is analytic in the whole complex plane. This is known for one-dimensional representations, the L-functions being then associated to Hecke characters — and in particular for Dirichlet L-functions. More generally Artin showed that the Artin conjecture is true for all representations induced from 1-dimensional representations. If the Galois group is supersolvable or more generally  monomial, then all representations are of this form so the Artin conjecture holds. André Weil proved the Artin conjecture in the case of function fields. Two-dimensional representations are classified by the nature of the image subgroup: it may be cyclic, dihedral, tetrahedral, octahedral, or icosahedral. The Artin conjecture for the cyclic or dihedral case follows easily from Erich Hecke's work. Langlands used the base change lifting to prove the tetrahedral case, and Jerrold Tunnell extended his work to cover the octahedral case; Andrew Wiles used these cases in his proof of the Taniyama–Shimura conjecture. Richard Taylor and  others have made some progress on the (non-solvable) icosahedral case; this is an active area of research. The Artin conjecture for odd, irreducible, two-dimensional representations follows from the proof of Serre's modularity conjecture, regardless of projective image subgroup. Brauer's theorem on induced characters implies that all Artin L-functions are products of positive and negative integral powers of Hecke L-functions, and are therefore meromorphic in the whole complex plane. The Dedekind conjecture. A weaker conjecture (sometimes known as Dedekind conjecture) states that if "M"/"K" is an extension of number fields, then the quotient formula_39 of  their Dedekind zeta functions is entire. The Aramata-Brauer theorem states that the conjecture holds if "M"/"K" is Galois. More generally, let "N" the Galois closure of "M" over "K", and "G" the Galois group of "N"/"K". The quotient formula_39 is equal to the Artin L-functions associated to the natural representation associated to the action of "G" on the "K"-invariants complex embedding of "M". Thus the Artin conjecture implies the Dedekind conjecture. The conjecture was proven when "G" is a solvable group, independently by Koji Uchida and R. W. van der Waall in 1975.