On the distribution of Ramanujan Sums over number fields

TL;DR

This paper extends the analysis of Ramanujan sums to quadratic and cubic number fields, providing new asymptotic formulas for their second moments and generalizing previous results over rationals.

Contribution

It introduces asymptotic formulas for second moments of Ramanujan sums over quadratic and cubic number fields, broadening the scope of earlier work over rationals.

Findings

Derived asymptotic formulas for second moments over quadratic fields

Extended results to cubic number fields

Obtained second moment results for specific number fields using domain properties

Abstract

For a number field $\mathbb{K}$, and integral ideals $\mathcal{I}$ and $\mathcal{J}$ in its number ring $\mathcal{O}_{\mathbb{K}}$, Nowak studied the asymptotic behaviour of the average of Ramanujan sums $C_{\mathcal{J}}({\mathcal{I}})$ over both ideals $\mathcal{I}$ and $\mathcal{J}$. In this article, we extend this investigation by establishing asymptotic formulas for the second moment of averages of Ramanujan sums over quadratic and cubic number fields, thereby generalizing previous works of Chen, Kumchev, Robles, and Roy on moments of averages of Ramanujan sums over rationals. Additionally, using a special property of certain integral domains, we obtain second moment results for Ramanujan sums over some other number fields.