A generalization of Graham’s estimate on the Barban–Vehov problem

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