Multiple zeta value

Definition
For $$ k_1, \ldots, k_r \in \mathbb{Z}_{>0} $$, the multiple zeta value is defined by

$$ \zeta(k_1,\ldots,k_r) = \sum_{0 < n_1 < \cdots < n_r} \frac{1}{n_1^{k_1} \cdots n_r^{k_r}} $$

For convergence we need $$ k_r > 1 $$.

Conventions/warning
There are two different conventions, related by reversing argument strings. When reading papers and results, be aware of which convention is being used. With the above convention

$$ \sum_{n_r > \cdots > n_1 > 0} \frac{1}{n_1^{k_1} \cdots n_r^{k_r}} = \zeta(k_r, \ldots, k_1) $$