  Suppose $\{ \lambda_d\}$ are Selberg's sieve weights and $1 \le w < y \le x$.
Graham's estimate on the Barban-Vehov problem shows that $\sum_{1 \le n \le x}
(\sum_{d|n} \lambda_d)^2 = \frac{x}{\log(y/w)} + O(\frac{x}{\log^2(y/w)})$. We
prove an analogue of this estimate for a sum over ideals of an arbitrary number
field $k$. Our asymptotic estimate remains the same; the only difference is
that the effective error term may depend on arithmetics of $k$. Our innovation
involves multiple counting results on ideals instead of integers. Notably, some
of the results are nontrivial generalizations. Furthermore, we prove a
corollary that leads to a new zero density estimate.

Number Theory