Many classical arithmetic functions can be thought of as functions on the set of (non-zero) ideals of $\mathbb{Z}$ rather than as functions on $\mathbb{N}$.

Example: For $n \in \mathbb{N}$ the divisor function $d(n)$ is defined to equal the number of divisors of $n$. Equivalently, we could define $d(I)$ to the number of ideals dividing the ideal $I \lhd \mathbb{Z}$, where we count once the fact that $(1)|(n)$.

Now, the associated Dirichlet could be written as $$ \sum^\infty_{n=1} d(n) n^{-s} = \sum_{I \lhd \mathbb{Z}} d(I) N(I)^{-s} = \zeta(s)^2. $$

If $K$ is a number field with ring of integers $\mathcal{O}$ we can extend the definition of $d(I)$ to the ideals of $\mathcal{O}$. Furthermore, we can consider the associated Dirichlet series: $$ \sum_{I \lhd \mathcal{O}} d(I) N(I)^{-s}. $$

QUESTIONS: Is this Dirichlet series equal to $\zeta_K(s)^2$, the square of the Dedekind zeta function of the field?

Does this phenomena generalize? ie if $a(n)$ is arithmetic function that can be equivalently defined on the ideals of $\mathbb{Z}$ and the Dirichlet series associated to $a(n)$ is a quotient of Riemann zeta functions (= Dedekind zeta function of $\mathbb{Q}$). Can we simply `replace' the Riemann zeta functions with the appropriate Dedekind zeta functions to obtain the Dirichet series associated to the extended a(n)?