On a generalization of arithmetic functions and the Ramanujan 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.

This paper has been withdrawn. The author noticed that this paper was not good