On Ramanujan Sums over a Dedekind Domain with Finite Norm Property

In this paper, we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property. We define the Ramanujan’s sums η(a, A) and η(B, A), where a is an arbitrary element in a Dedekind domain, B is an ideal and A is a non-zero ideal. In particular, we discuss the Kluyver formula and Hèolder formula for η(a, A) and η(B, A). We also prove the reciprocity formula enjoyed byη(B, A) and the orthogonality relations for η(a, A) in the last two parts.