Moments in the Chebotarev density theorem: non-Gaussian families

In this paper we investigate higher moments attached to the Chebotarev Density Theorem. Our focus is on the impact that peculiar Galois group structures have on the limiting distribution. Precisely we consider in this paper the case of groups having a character of large degree. Under the Generalized Riemann Hypothesis, we prove in particular that there exists families of Galois extensions of number fields having doubly transitive Frobenius group for which no Gaussian limiting distribution occurs.