Consider the sum of divisor function $$ \sigma_0(n) = \sum_{d\mid n} 1. $$ This is known to satisfy $\sum_{n\leq x} \sigma_0(n) = (x\log x)+2\gamma x+\mathcal{O}(\sqrt{x})$. If, instead, we shift the setting to a number field $K/\mathbb{Q}$ and consider the analogous sum $$ \sigma_0(\mathfrak{n}) =\sum_{\mathfrak{d}\mid \mathfrak{n}} N_{K/\mathbb{Q}}(\mathfrak{d}), $$ what can be said about the sum $$ \sum_{N_{K/\mathbb{Q}}(\mathfrak{n})\leq x} \sigma_0(\mathfrak{n})? $$ What about the related function $$f(n)=\sum_{d\mid n} 2^{\omega(d)}$$ where $\omega(n)$ is the number of prime divisors of $n$, but considered in the context of a number field $K$. In particular, what's the main term of $$ \sum_{N_{K/\mathbb{Q}}(\mathfrak{n})\leq x}\sum_{\mathfrak{d}\mid \mathfrak{n}} 2^{\omega(\mathfrak{d})} $$ I'd be happy with a reference.
Edit: I accidentally started with $\sigma_1$ when I meant $\sigma_0$.
Edit 2: In response to @sea turtles comment, what if we restrict to a totally real field?