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 n S 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 n S 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.