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 ${\mathfrak{F}}_k^G(Q)$ be the family of number fields $K$ with absolute discriminant $D_K$ at most $Q$ such that $K\mathrm{/}k$ 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 $K\in {\mathfrak{F}}_k^G(Q)$ with at most $O_{ϵ}(Q^{ϵ})$ exceptions, the $L$-functions associated to the faithful Artin representations of $\mathrm{G}\mathrm{a}\mathrm{l}(K\mathrm{/}k)$ 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:

1. there exist infinitely many degree $n$ $S_n$-fields over $\mathbb{Q}$ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke;
2. for a prime $p$, the periodic torus orbits attached to the ideal classes of almost all totally real degree $p$ fields $F$ over $\mathbb{Q}$ equidistribute on ${\mathrm{P}\mathrm{G}\mathrm{L}}_p(\mathbb{Z})\mathrm{\backslash }{\mathrm{P}\mathrm{G}\mathrm{L}}_p(\mathbb{R})$ with respect to Haar measure;
3. for each $\mathrm{\ell }\ge 2$, the $\mathrm{\ell }$-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
4. an effective variant of the Chebotarev density theorem holds for almost all fields in such families.