A Barban-Davenport-Halberstam asymptotic for number fields

TL;DR

This paper extends the Barban-Davenport-Halberstam theorem to number fields, providing an asymptotic formula for the distribution of prime ideals with specific norm properties, improving previous mean square error estimates.

Contribution

It establishes an asymptotic formula for prime ideal distributions in Galois number fields, generalizing classical results to a broader algebraic setting.

Findings

Derived an asymptotic formula for prime ideal distribution in Galois number fields.

Connected the result to the classical Barban-Davenport-Halberstam theorem.

Enhanced understanding of prime ideal distribution with explicit asymptotics.

Abstract

Let $K$ be a fixed number field, and assume that $K$ is Galois over $\qq$. Previously, the author showed that when estimating the number of prime ideals with norm congruent to $a$ modulo $q$ via the Chebotar\"ev Density Theorem, the mean square error in the approximation is small when averaging over all $q\le Q$ and all appropriate $a$. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the case $K=\qq$.