The average size of Ramanujan sums over quadratic number fields(II)

Abstract

In this paper we study Ramanujan sums $c_{\mathbf{m}}(\mathbf{n})$, where $\mathbf{m}$ and $\mathbf{n}$ are integral ideals in an arbitrary quadratic number field. We give some new results about the asymptotic behavior of sums of $c_{\mathbf{m}}(\mathbf{n})$ over both $\mathbf{m}$ and $\mathbf{n}$.