A supplement to Chebotarev's density theorem

TL;DR

This paper extends Chebotarev's density theorem by establishing a link between the distribution of prime ideals with specific Frobenius conjugacy classes and the density of totally split primes, using properties of zeta functions.

Contribution

It provides a new result connecting the density of primes with specific Frobenius classes to the density of totally split primes under a power-saving error term.

Findings

Density of primes with Frobenius in a conjugacy class tends to |C|/|G|

Relation between poles of Dirichlet series and zeros of zeta functions

Power saving error term in prime distribution

Abstract

Let $L/K$ be a Galois extension of number fields with Galois group $G$. We show that if the density of prime ideals in $K$ that split totally in $L$ tends to $1/|G|$ with a power saving error term, then the density of prime ideals in $K$ whose Frobenius is a given conjugacy class $C\subset G$ tends to $|C|/|G|$ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of $\zeta_L(s)/\zeta_K(s)$.