We prove a new effective Chebotarev density theorem for Galois extensions L/Q that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of L); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of L, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal L-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal L-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of L-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for -torsion in class groups, for all integers ≥ 1, applicable to infinite families of fields of arbitrarily large degree.

In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple Hpxq " tgx`h j u k j"1 of linear forms in Zrxs, the set Hpnq " tgn`h j u k j"1 contains at least m primes for infinitely many n P N. In this note, we deduce that Hpnq " tgn`h j u k j"1 contains at least m consecutive primes for infinitely many n P N. We answer an old question of Erdős and Turán by producing strings of m`1 consecutive primes whose successive gaps δ 1 , . . . , δ m form an increasing (resp. decreasing) sequence. We also show that such strings exist with δ j´1 | δ j for 2 ď j ď m. For any coprime integers a and D we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class a mod D.

An effective Chebotarev density theorem for a fixed normal extension L/Q provides an asymptotic, with an explicit error term, for the number of primes of bounded size with a prescribed splitting type in L. In many applications one is most interested in the case where the primes are small (with respect to the absolute discriminant of L); this is well-known to be closely related to the Generalized Riemann Hypothesis for the Dedekind zeta function of L. In this work we prove a new effective Chebotarev density theorem, independent of GRH, that improves the previously known unconditional error term and allows primes to be taken quite small (certainly as small as an arbitrarily small power of the discriminant of L); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Such a family has fixed degree, fixed Galois group of the Galois closure, and in certain cases a ramification restriction on all tamely ramified primes in each field; examples include totally ramified cyclic fields, degree n Sn-fields with square-free discriminant, and degree n An-fields. In all cases, our work is independent of GRH; in some cases we assume the strong Artin conjecture or hypotheses on counting number fields.The technical innovation leading to our main theorem is a new idea to extend a result of Kowalski and Michel, a priori for the average density of zeroes of a family of cuspidal L-functions, to a family of non-cuspidal L-functions. Unexpectedly, a crucial step in this extension relies on counting the number of fields within a certain family, with fixed discriminant. Finally, in order to show that comparatively few fields could possibly be exceptions to our main theorem (under GRH, none are exceptions), we must obtain lower bounds for the number of fields within a certain family, with bounded discriminant. We prove in particular the first lower bound for the number of degree n An-fields with bounded discriminant.The new effective Chebotarev theorem is expected to have many applications, of which we demonstrate two. First we prove (for all integers ≥ 1) nontrivial bounds for -torsion in the class groups of "almost all" fields in the families of fields we consider. This provides the first nontrivial upper bounds for -torsion, for all integers ≥ 1, applicable to infinite families of fields of arbitrarily large degree. Second, in answer to a question of Ruppert, we prove that within each family, "almost all" fields have a small generator.

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