Q:

What's a good symbol to represent many summations?  Can a tensor product glyph be used? 

What's a good symbol for many repeated summations?  I vaguely remember seeing something like 
$$
\otimes_{j=1}^N \sum_{n_j=-\infty}^\infty f(\vec{n})= \sum_{n_1=-\infty}^\infty\sum_{n_2=-\infty}^\infty\cdots\sum_{n_N=-\infty}^\infty f(\vec{n}).$$ 
Is this used in practice?  If not, is there a commonly accepted shorthand? 

A:

You essentially want to sum over all vectors of length $N$ with integer entries.  If we say $\mathcal{Z} = \otimes^N_{j=1} \mathbb{Z}$, then you can write the above sum compactly as:
 $$\sum_{\vec{z} \in \mathcal{Z}} f(\vec{z})$$

