Abstract

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.

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