Formally, what is the meaning of sigma notation with two variables in the subscript and one variable in the superscipt?

Question

For example, what would $\displaystyle \sum _{i,j=1} ^n a_i a_j$ mean?

Answer 1

It means $$ \sum_{i=1}^n\sum_{j=1}^n a_ia_j $$ that is both indices range from $1$ to $n$.

In this particular case, you can even simplify it further as $$ \sum_{i=1}^n\sum_{j=1}^n a_ia_j = \sum_{i=1}^na_i\sum_{j=1}^n a_j = \left(\sum_{j=1}^n a_i\right)^2 $$ since the term $a_ia_j$ can be decoupled between $i$ and $j$.