An effective version of Chebotarev's density theorem

TL;DR

This paper refines the effective Chebotarev density theorem by providing explicit constants and sharper bounds for small degree extensions, utilizing advanced zero-free regions and zero estimates for Dedekind zeta functions.

Contribution

It offers an explicit refinement applicable to all non-rational fields with improved bounds and explicit constants, extending prior results by Lagarias and Odlyzko.

Findings

Explicit formula for smoothed prime ideal counting function

Sharper bounds for small degree extensions

Explicit constants in the density theorem

Abstract

Chebotarev's density theorem asserts that the prime ideals are equidistributed among the conjugacy classes of the Galois group of any normal extension of number fields. An effective version of this theorem was first established by Lagarias and Odlyzko in 1977. In this article, we present an explicit refinement of their statement that applies to all non-rational fields, with every implicit constant expressed explicitly in terms of the field invariants. Additionally, we provide a sharper bound for extensions of sufficiently small degree. Our approach begins by proving an explicit formula for a smoothed prime ideal counting function. This relies on recent zero-free regions for Dedekind zeta functions, improved estimates on the number of low-lying zeros, and precise bounds for sums over the non-trivial zeros of the Dedekind $\zeta$-function.