A Barban-Davenport-Halberstam asymptotic for number fields

Abstract

Let K K be a fixed number field, and assume that K K is Galois over Q \mathbb {Q} . Previously, the author showed that when estimating the number of prime ideals with norm congruent to a a modulo q q via the Chebotarëv Density Theorem, the mean square error in the approximation is small when averaging over all q ≤ Q q\le Q and all appropriate a 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 = Q K=\mathbb {Q} .