A new bound on the relative error in the Chebotarev density theorem

TL;DR

This paper improves the uniformity bounds in the Chebotarev density theorem for Galois extensions using nonabelian base change, leading to a better estimate for the least norm of unramified primes with a given Artin symbol.

Contribution

It provides the first theoretical improvement over Weiss's bound for the least norm of unramified primes with a specified Artin symbol.

Findings

Unconditional improvement of Chebotarev density bounds

First theoretical enhancement over Weiss's bound

Better estimates for least unramified prime norms

Abstract

We unconditionally improve the uniformity in the Chebotarev density theorem for Galois extensions of number fields using nonabelian base change. This leads to the first theoretical improvement over Weiss's bound for the least norm of an unramified prime ideal with given Artin symbol.