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We prove a new effective Chebotarev density theorem for Galois extensions$L/\mathbb{Q}$ that allows one to count small primes (even as small as anarbitrarily small power of the discriminant of $L$); this theorem holds for theGalois closures of "almost all" number fields that lie in an appropriate familyof field extensions. Previously, applying Chebotarev in such small rangesrequired assuming the Generalized Riemann Hypothesis. The error term in thisnew Chebotarev density theorem also avoids the effect of an exceptional zero ofthe Dedekind zeta function of $L$, without assuming GRH. We give many different"appropriate families," including families of arbitrarily large degree. To dothis, we first prove a new effective Chebotarev density theorem that requires azero-free region of the Dedekind zeta function. Then we prove that almost allnumber fields in our families yield such a zero-free region. The innovationthat allows us to achieve this is a delicate new method for controlling zeroesof certain families of non-cuspidal $L$-functions. This builds on, and greatlygeneralizes the applicability of, work of Kowalski and Michel on the averagedensity of zeroes of a family of cuspidal $L$-functions. A surprising featureof this new method, which we expect will have independent interest, is that wecontrol the number of zeroes in the family of $L$-functions by bounding thenumber of certain associated fields with fixed discriminant. As an applicationof the new Chebotarev density theorem, we prove the first nontrivial upperbounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$,applicable to infinite families of fields of arbitrarily large degree.

An effective Chebotarev density theorem for families of number fields, with an application to $\ell$-torsion in class groups