TITLE: Can one define "Ramanujan Summation" over algebraic number fields?
QUESTION [9 upvotes]: With some trepidation, I ask to "evaluate" badly divergent sums.  Generalizing $\sum n = -\tfrac{1}{12}$ what would be the value of this sum over $\mathbb{Z}[i]$?
$$\sum_{m,n \geq 0} (m+in) \hspace{0.25in}\text{and}\hspace{0.25in}\sum_{0 < m < n } (m+in)$$
This cursory and highly dangerous check shows that we would probably need higher order terms in the Euler McLarurin expansion.
$$  \sum_{m,n \geq 0} (m+in)  = \sum_{m,n \geq 0} m + i \sum_{m,n \geq 0}n 
=  \sum_{n \geq 0} -\tfrac{1}{12} + i \sum_{m \geq 0} -\tfrac{1}{12} = ??? $$
In a sense, all Ramanujan sums do is subtract the sum from an approximating integeral.  I tried:
$$ \sum_{m,n \geq 0}^{M,N} (m+in)  - \int_0^M \int_0^N (x + i y)\,dx \, dy
 = \tfrac{MN(M+N+2)}{2} - \tfrac{MN(M+N)}{2} = \tfrac{MN}{2}$$
This doesn't look right at all, but I did find something called the Khovanskii-Puhlikov theorem which looks like it might give the answer.

In general for any number field $K$ and lattice cone $V \subseteq \mathcal{O}_K$ in the ring of integers we could try to define:
$$ \sum_{x \in V \subset \mathcal{O}_K} x $$
So I wonder if someone has attempted to define such summations and what they evaluate to.

REPLY [6 votes]: I don't think this is exactly what you are looking for, but it seems too related not to mention. You can generalize zeta regularization to any number field $K$, although you run into an open problem.
Over $\mathbb{Q}$ we have the classic (Riemann) zeta regularization $\sum n=\zeta(-1)$, the right hand side of which can be defined by analytic continuation and equals the (in)famous $-1/12$.
Now, for an arbitrary number field $K$, take its Dedekind zeta function, which is also a meromorphic function with a simple pole on 1, and then define the sum over some (see Gerry's comment below) integral ideals of $K$:
$$\sum_{\mathfrak{a}' \in \mathcal{O}_K} \mathfrak{N(a)}=\zeta_K(-1)$$
The exact value of this sum is not know in general. The better understood case is for $K$ totally real, in which it is known to be an rational number (Klingen-Siegel) and conjectured to be related to algebraic K-groups (Birch-Tate).