I know that: $\sum\_{n\in \mathbb N} \frac{1 }{n^{p}}$ converges if $p>1$ and diverges if $p\leq 1$

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> **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? 
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