A unified and improved Chebotarev density theorem

TL;DR

This paper proves an improved, unconditional effective Chebotarev density theorem that enhances prime counting in number fields, especially with small primes and in the presence of Landau-Siegel zeros, offering new asymptotic formulas and applications.

Contribution

It introduces a unified, improved Chebotarev density theorem with uniform bounds and novel applications, including prime counts for quadratic forms with coprime conditions.

Findings

Enhanced bounds for the least prime ideal with a given Artin symbol

New asymptotic formulas for counting primes in number fields

Effective prime counting for quadratic forms with coprimality constraints

Abstract

We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau-Siegel zero is present. Our main theorem interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun-Titchmarsh theorem proved by the authors. We also present a new application of our main result that exhibits considerable gains over earlier versions of the Chebotarev density theorem. If $f$ is a positive definite primitive binary quadratic form then we count lattice points $(u,v) \in \mathbb{Z}^2$ such that $f(u,v)$ is prime and $u, v$ have no prime factors $\leq z$ with uniformity in $z$ and the discriminant of $f$.