On a generalization of arithmetic functions and the Ramanujan sums

TL;DR

This paper extends the theory of arithmetic functions and Ramanujan sums from the integers to algebraic number fields, establishing foundational properties in this broader context.

Contribution

It introduces a generalization of arithmetic functions and Ramanujan sums over number fields, expanding classical concepts to a more general algebraic setting.

Findings

Generalized properties of arithmetic functions over number fields

Defined and analyzed properties of generalized Ramanujan sums over $K$

Established foundational results for these generalized sums

Abstract

Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define the generalized Ramanujan sums over $K$ and show some properties.