A Generalization of Graham's Estimate on the Barban-Vehov Problem

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 ${\sum }_{1\le n\le x}({\sum }_{d\mathrm{\mid }n}{\lambda }_d{)}^2=\frac{x}{\log (y\mathrm{/}w)}+O(\frac{x}{{\log }^2(y\mathrm{/}w)})$. 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.