A generalization of the Barban-Davenport-Halberstam Theorem to number fields

TL;DR

This paper extends the Barban-Davenport-Halberstam Theorem to number fields, providing bounds on the mean square error in prime counting estimates using large sieve inequalities.

Contribution

It generalizes a classical prime distribution result to the setting of number fields, employing large sieve techniques for the analysis.

Findings

Derived an upper bound analogous to the classical theorem

Applied large sieve inequality in the context of number fields

Enhanced understanding of prime distribution in algebraic number fields

Abstract

For a fixed number field $K$, we consider the mean square error in estimating the number of primes with norm congruent to $a$ modulo $q$ by the Chebotar\"ev Density Theorem when averaging over all $q\le Q$ and all appropriate $a$. Using a large sieve inequality, we obtain an upper bound similar to the Barban-Davenport-Halberstam Theorem.