Usual techniques (hyperbola method, moving contour, etc.) yield the following bounds $$\sum_{d < x} d^r = \frac{1}{r+1}x^{r+1} + O(x^r)$$ or $$\sum_{d < x} \frac{\mu(d)}{d^2} = \frac{1}{\zeta(2)} + O(x^{-1}).$$
I am interested with what happens when the classical setting is replaced by a number field $F$ of degree $n$ over $\mathbb{Q}$. How can we obtain the analogous bounds for $$\sum_{Nd < x} Nd^r = ? $$ or $$\sum_{Nd < x} \frac{\mu(d)}{Nd^2} = ?$$
where the sums are over integer ideals $d$ of $F$ and $Nd$ is the norm of $d$. I saw in particular (without proof) the statement $$\sum_{Nd < x} Nd^r = \frac{\zeta^\star(1)}{r+1} X^{r+1} + O(X^{r + 1 - 2/(n+1)}) $$
where $\zeta^\star(1)$ is the residue of the Dedekind zeta function of $F$ at $1$. How are these statement proven? Is there any reference for "classical analytic number theory" matters like this one for more general number fields?