How to interpret $\sum_{n\in \mathbb N^{d}} \frac{1}{n^{p}};$ and when it is converges? I know that: $\sum_{n\in \mathbb N} \frac{1 }{n^{p}}$ converges if $p>1$ and diverges if $p\leq 1$

My Question is: What is an analogue  this in  more than one variable (say $d$)? Does it make sense to talk of $\sum_{n\in \mathbb N^{d}} \frac{1 }{n^{p}}$? (How I should interpretative  $\frac{1}{n^{p}}$ if $n\in \mathbb N^{d}$, I mean: is there any standard meaning..., etc...) After giving this meaning, what about its convergence? 

 A: I think it is very natural to consider summing across integer points of a lattice as the multidimensional analogue. So for instance, you might consider
$$ \sum_{\vec n \in \mathbb{N}^d} \frac{1}{\lvert \vec n \rvert^p}$$
where $\lvert \vec n \rvert$ is the length of the vector $\vec n$. This agrees with the normal sum when $d = 1$.
And in general, it's not so hard to show that this converges for $p > d$.
Alternately, it might be natural to consider
$$ \sum_{n_i \in \mathbb{N}} \frac{1}{n_1^{p_1} n_2^{p_2} \cdots n_k^{p_k}}$$
as each of the $p_i$ vary. This will converge for $p_i > 1$.
A: For the question to make sense you can replace $n^p$ with $||(n_1,n_2,\ldots,n_d)||^p$, where $||\cdot||$ is the Euclidean norm of the vector. The number of $d$-dimensional vectors of norm in an interval of the form $[M,M+1]$ is well approximated by $K\cdot M^{d-1}$, where $K$ is a constant depending on $d$. Think: a shell of thickness $1$ around a $d$-dimensional sphere of radius $M$.
It follows easily that the series converges, if $p>d$.

This type of thinking (with $d=2$) appears in the construction of doubly periodic meromorphic functions on the complex plane. The derivative of Weierstrass $\wp$-function with poles at points determined by a lattice is done as such a summation with $p=3$. The summation is (more naturally) over $\Bbb{Z}^2$ rather than $\Bbb{N}^2$. It follows that the double sum will converge absolutely and uniformly in any compact set allowing us to conclude both meromorphicity and double periodicity.
