what does this summation mean?

Question

I apologize if this question has been asked. I know several similar ones have been asked but I cannot find one answering this in particular.

I want to know what this summation means: $$\sum_{i,j=1}^{M}$$
Is this equivalent to $$\sum_{i=1}^M \sum_{j=1}^M$$
or does it mean that i and j will be incremented together at the same time?
If context is needed, the equation is coming from here equation 16 on the third page (page 392 of the journal).

It's the sum over the set of indices $(i,j)$ with $1\leqslant i,j\leqslant M$ which is equivalent to $\sum_{i=1}^M\sum_{j=1}^M$. — Stefan Hansen, Nov 18 '14 at 13:05
$\sum_{i,j=1}^{M}\equiv \sum_{i=1}^M \sum_{j=1}^M$ with no doubt. — Raúl Alegre, Nov 18 '14 at 13:06

score 1 · Accepted Answer · answered Nov 18 '14 at 13:06

We have

$$\sum_{i,j=1}^{M}a_{i,j}=\sum_{1\le i,j\le M}a_{i,j}=\sum_{i=1}^M\sum_{j=1}^Ma_{i,j}=\sum_{j=1}^M\sum_{i=1}^Ma_{i,j}$$ means that we sum the terms $a_{i,j}$ with all the possible values of $i$ and $j$ between $1$ and $M$.

what does this summation mean?

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