An approximate form of Artin’s holomorphy conjecture and non-vanishing of Artin L -functions

Let k be a number field and G be a finite group. Let F _k ^G(Q) be the family of number fields K with absolute discriminant D _K at most Q such that K/k is normal with Galois group isomorphic to G. If G is the symmetric group S _n or any transitive group of prime degree, then we unconditionally prove that for all K∈F _k ^G(Q) with at most O _ε(Q ^ε) exceptions, the L-functions associated to the faithful Artin representations of Gal(K/k) have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that: there exist infinitely many degree nS _n-fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke; for a prime p, the periodic torus orbits attached to the ideal classes of almost all totally real degree p fields F over ℚ equidistribute on PGL _p(Z)∖PGL _p(R) with respect to Haar measure; for each ℓ≥2, the ℓ-torsion subgroups of the ideal class groups of almost all degree p fields over k (resp. almost all degree nS _n-fields over k) are as small as GRH implies; and an effective variant of the Chebotarev density theorem holds for almost all fields in such families.