An effective Chebotarev density theorem for a fixed normal extension $L/\mathbb{Q}$ 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 tame 查看全文>>