Generalization of the Mobius inversion formula to Dedekind domains

Question

Since Dedekind domains have the same factorization theorems for ideals analogous to that of the integers, there is some generalization of the Mobius inversion formula to Dedekind domains?

Answer 1

Möbius inversion is most easily generalized by using

$$\mu_K:\begin{cases} I(K)\to \{0,\pm 1\} \\ \mu(\mathfrak{p}_1\ldots \mathfrak{p})r=(-1)^r & \mathfrak{p_i}+\mathfrak{p_j}=\mathcal{O}_K \;(i\ne j) \\ \mu(\mathfrak{a})=0 & \mathfrak{p}^2|\mathfrak{a}\end{cases}$$

Then the statement is just that

$${1\over\zeta_K(s)}=\sum_{I}\mu(I)N(I)^{-s}.$$

This formula readily follows from the Euler product and the unique factorization of ideals in a Dedekind domain.