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

TL;DR

This paper investigates the asymptotic behavior of Ramanujan sums over quadratic number fields, extending previous work by analyzing sums over integral ideals in these fields.

Contribution

It provides new results on the asymptotic behavior of Ramanujan sums over quadratic number fields, generalizing classical results to ideal sums.

Findings

Derived asymptotic formulas for sums of Ramanujan sums over ideals

Extended classical results to quadratic number fields

Enhanced understanding of Ramanujan sums in algebraic number theory

Abstract

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