A variant of the Barban-Davenport-Halberstam Theorem

TL;DR

This paper extends the classical Barban-Davenport-Halberstam Theorem to Galois extensions of number fields, showing that the average squared error term in counting prime ideals with fixed Frobenius class and congruence conditions is small.

Contribution

It introduces a variation of the Barban-Davenport-Halberstam Theorem applicable to prime ideals in Galois extensions, focusing on error term averages.

Findings

The squared error term is small on average for the problem considered.

The result generalizes the classical theorem to number fields with Frobenius class conditions.

Provides bounds on the error term in the Chebotar"ev Density Theorem context.

Abstract

Let $L/K$ be a Galois extension of number fields. The problem of counting the number of prime ideals $\mathfrak p$ of $K$ with fixed Frobenius class in $\mathrm{Gal}(L/K)$ and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotar\"ev Density Theorem for this problem is small "on average." The result may be viewed as a variation on the classical Barban-Davenport-Halberstam Theorem.