Multiple zeta function

In mathematics, the multiple zeta values (MZVs) are sums over simplicies. They are defined by

$\zeta (a_{1},\ldots ,a_{k})=\sum _{n_{1}>n_{2}>\dots >n_{k}>0}\ {\frac {1}{n_{1}^{a_{1}}\dots n_{k}^{a_{k}}}}=\sum _{n_{1}>n_{2}>\dots >n_{k}>0}\ \prod _{i=1}^{k}{\frac {1}{n_{i}^{a_{i}}}}.\!$

The standard shorthand for writing multiple zeta values is to place repeating of the argument within braces and use a superscript to indicate the number of repetitions. For example, $\zeta (2,1,2,1,3)=\zeta (\{2,1\}^{2},3)$ and $\zeta (2,1,1,3,1,1)=\zeta (2,\{1\}^{2},3,\{1\}^{2}).$