Moments in the Chebotarev density theorem: non-Gaussian families

TL;DR

This paper explores how specific Galois group structures influence the distribution of primes in number fields, showing that certain families do not follow Gaussian limits under the Generalized Riemann Hypothesis.

Contribution

It demonstrates that families of Galois extensions with particular group structures can exhibit non-Gaussian limiting distributions, extending understanding of Chebotarev density moments.

Findings

Existence of Galois extension families with non-Gaussian limits

Impact of large degree characters on distribution

Conditional results under GRH

Abstract

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.