The Average Size of Ramanujan Sums over Cubic Number Fields

TL;DR

This paper investigates the average behavior of Ramanujan sums over cubic number fields, analyzing their asymptotic properties when summed over integral ideals.

Contribution

It provides new insights into the asymptotic behavior of Ramanujan sums in cubic number fields, extending classical results to this setting.

Findings

Derived asymptotic formulas for sums of Ramanujan sums over ideals

Established bounds for the average size of Ramanujan sums in cubic fields

Extended classical number theory results to cubic number field context

Abstract

Let K be a cubic number field. In this paper, we study the Ramanujan sums c_{J}(I), where I and J are integral ideals in O_{K}. The asymptotic behaviour of sums of c_{J}(I) over both I and J is investigated.