An explicit Chebotarev Density Theorem on average and applications

The study of specific families of polynomials and their splitting fields can provide useful examples and evidences for conjectures regarding invariants related to number field extensions and related objects. The main example that can illustrate the relevance of the work, is the family P0 n,N of degree n monic polynomials f with integer coefficients, so that the maximum of the absolute values of the coefficients is less or equal than N, and the splitting field Kf over the rationals Q is the full symmetric group Sn. We let N → +∞. With a little work, this can actually be generalized to polynomials with integral coefficients in a fixed number field of degree d over Q. Let π_f,r(x) be the function counting the primes less or equal than x such that f ∈ P0n,N has a fixed square-free splitting type r modulo p. It turns out that the quantity (π_f,r(x)−δ(r)π(x)((δ(r)−δ(r)2)π(x))^(−1/2) is distributed like a normal distribution with mean 0 and variance 1, whenever x is small compared to N, e.g. x = N1/ log logN. Here δ(r) is the coefficient in the asymptotic predicted by the classical Chebotarev theorem. This result leads to interesting applications, as finding upper bounds for the torsion part of the class number in terms of the absolute discriminant, as it was done for other infinite families by Ellenberg, Venkatesh and Heath-Brown, Pierce.