I am working on generalizing some works from the usual rational case to general number fields. That implies some technical changes I am not really at ease with. For instance:
$$\sum_{m \leqslant X} m = \frac{X(X-1)}{2} = \frac{X^2}{2} + O(X)$$
Generalizing it, I end with the following sum to estimate, where $\mathfrak{m}$ stands for ideals of the integer ring $\mathcal{O}_F$ (analogues of the integers in the case of $\mathbf{Q}$) and $N$ is the norm on $F$, and I would like to prove this kind of estimate:
$$\sum_{N\mathfrak{m} \leqslant X} N\mathfrak{m} = \zeta_F(-1) \frac{X^2}{2} + O(X)$$
where $\zeta_F(-1)$ stands for the value of the zêta function of $F$ at -1. Is it true, and how can I prove this kind of results (more than formally saying something like $\zeta(-1) = \sum N\mathfrak{m}$ ?
Thanks in advance for any clue or idea ;)