A Barban-Davenport-Halberstam asymptotic for number fields

Let $K$ be a fixed number field, and assume that $K$ is Galois over $\text{qq}$. 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 $q\le Q$andallappropriate$a$. 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 $K=\text{qq}$.