It is well known that the function $$\sigma_k(n)=\sum_{d|n}d^k$$ has a generating function.
For a number field $K$, suppose that $\mathfrak{a}, \mathfrak{b}$ are ideals in some ideal class $C$ and let $N$ denote the norm of the ideal. Does the function $$\sigma_{k,\mathfrak{a}}=\sum_{\mathfrak{b}|\mathfrak{a}}N(\mathfrak{b})^k$$ have a generating function?
Not really. Let $$\zeta_K(s) = \sum_{I \in O_K} N(I)^{-s}$$ The problem is that there is no a bijective correspondence between $I$ and $N(I)$, thus we don't have $$\zeta_K(s)\zeta_K(s-k) = \sum_{I \in O_K} N(I)^{-s} \sum_{J\in O_K,J\supset I} N(J)^k$$ as in the case $K = \Bbb{Q}$.
In particular the number of ideals dividing the ideals $(2+i)^2$ and $(2+i)(2-i)$ of $\Bbb{Z}[i]$ are not the same while their norm is equal.
To repair it you'd something like this : for simplicity let $K = \Bbb{Q}(i)$ and $$F(s) = \frac{1}{1-\scriptstyle\pmatrix{2^{-s}& 0 \\ 0 & 0}}\prod_{p \equiv 1 \bmod 4} \frac{1}{(1-\scriptstyle\pmatrix{p^{-s}& 0 \\ 0 & 0})(1-\pmatrix{0& 0 \\ 0 & p^{-s}})} \prod_{p \equiv 3 \bmod 4} \frac{1}{1-\scriptstyle\pmatrix{p^{-2s}& 0 \\ 0 & 0}}$$
$$G(s) = \frac{1}{1-2^k\scriptstyle\pmatrix{2^{-s}& 0 \\ 0 & 0}}\prod_{p \equiv 1 \bmod 4} \frac{1}{(1-p^k\scriptstyle\pmatrix{p^{-s}& 0 \\ 0 & 0})(1-p^k\scriptstyle\pmatrix{0 & 0 \\ 0 & p^{-s}})} \prod_{p \equiv 3 \bmod 4} \frac{1}{1-p^{2k}\scriptstyle\pmatrix{p^{-2s}& 0 \\ 0 & 0}}$$
Then $$F(s)G(s) = \sum_{A \in M} A^{-s} \sum_{B \in M, B | A} \det(A)^k$$ where $M$ is the monoid of diagonal matrices $\pmatrix{c & 0 \\ 0 & d}$ where $c,d$ are norms of ideals in $O_K= \Bbb {Z}[i]$ and $p | d \implies p \equiv 1 \bmod 4$, $B| A$ means divisibility in the sense of that monoid, each $A \in M$ represents a unique ideal $I \in O_K$ and $$\sum_{B \in M, B | A} \det(A)^k = \sum_{J\in O_K, J \supset I} N(J)^k$$ In general those $F(s),G(s)$ can be defined in term of Artin L-functions of $K$ thus the PNT estimates still apply to $\sum_{J\in O_K, J \supset I} N(J)^k$.