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

TL;DR

This paper extends Graham's estimate for the Barban-Vehov problem from integers to ideals in arbitrary number fields, incorporating the field's arithmetic into the error term and introducing novel counting techniques.

Contribution

It generalizes Graham's estimate to ideals in number fields, using new multiple counting results and deriving a zero density estimate.

Findings

Asymptotic estimate remains the same for number fields

Effective error term depends on number field arithmetic

New zero density estimate derived from the generalization

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.