The sum converges for all positive integer arguments, except when the first argument equals one, for instance as in MultiZeta(1,2,3), in which case the function diverges.

With no arguments, MultiZeta() is defined as equal to 1.

For one argument, MultiZeta reduces to the Riemann Zeta function:

$\mathrm{\%MultiZeta}\left(43\right)\=\mathrm{MultiZeta}\left(43\right)$

$\mathrm{MultiZeta}\left(43\right)=\mathrm{\zeta }\left(43\right)$

$\left(\mathrm{\%MultiZeta}\=\mathrm{MultiZeta}\right)\left(27\,27\right)$

$\mathrm{MultiZeta}\left(27\,27\right)=\frac{{\mathrm{\zeta }\left(27\right)}^2}{2}-\frac{\mathrm{\zeta }\left(54\right)}{2}$

$\left(\mathrm{\%MultiZeta}\=\mathrm{MultiZeta}\right)\left(11\,8\right)\;$

$\mathrm{MultiZeta}\left(11\,8\right)=-\frac{75583\mathrm{\zeta }\left(19\right)}{2}+\frac{9724{\mathrm{\pi }}^2\mathrm{\zeta }\left(17\right)}{3}+\frac{4433{\mathrm{\pi }}^4\mathrm{\zeta }\left(15\right)}{90}+\frac{286{\mathrm{\pi }}^6\mathrm{\zeta }\left(13\right)}{315}+\frac{121{\mathrm{\pi }}^8\mathrm{\zeta }\left(11\right)}{9450}+\frac{8{\mathrm{\pi }}^{10}\mathrm{\zeta }\left(9\right)}{93555}$

$\left(\mathrm{\%MultiZeta}\=\mathrm{MultiZeta}\right)\left(2\,1\,4\right)$

$\mathrm{MultiZeta}\left(2\,1\,4\right)=\frac{7{\mathrm{\pi }}^4\mathrm{\zeta }\left(3\right)}{360}-\frac{11{\mathrm{\pi }}^2\mathrm{\zeta }\left(5\right)}{12}+\frac{61\mathrm{\zeta }\left(7\right)}{8}$

$\left(\mathrm{\%MultiPolylog}\=\mathrm{MultiPolylog}\right)\left(\left[2\,3\,4\,5\right]\,\left[1\,1\,1\,1\right]\right)\;$

$\mathrm{MultiPolylog}\left(\left[2\,3\,4\,5\right]\,\left[1\,1\,1\,1\right]\right)=\mathrm{MultiZeta}\left(2\,3\,4\,5\right)$

$\mathrm{MultiZeta}\left(7\,9\right)\mathrm{MultiZeta}\left(6\right)$

$\frac{\mathrm{MultiZeta}\left(7\,9\right){\mathrm{\pi }}^6}{945}$

$\mathrm{MultiZeta}\left(7\,9\,6\right)+\mathrm{MultiZeta}\left(7\,6\,9\right)+\mathrm{MultiZeta}\left(6\,7\,9\right)+\mathrm{MultiZeta}\left(13\,9\right)+\mathrm{MultiZeta}\left(7\,15\right)$

$\mathrm{MultiZeta}\left(7\,9\,6\right)+\mathrm{MultiZeta}\left(7\,6\,9\right)+\mathrm{MultiZeta}\left(6\,7\,9\right)+\mathrm{MultiZeta}\left(13\,9\right)+\mathrm{MultiZeta}\left(7\,15\right)$

$\mathrm{evalf}\left[5\right]\left(\=\right)$

$0.0084952=0.0084952$

$\mathrm{MultiZeta}\left(2\,3\,4\right)$

$\mathrm{MultiZeta}\left(2\,3\,4\right)$

$\mathrm{MultiZeta}\left(2\,1\,1\,2\,1\,2\right)$

$\mathrm{MultiZeta}\left(2\,1\,1\,2\,1\,2\right)$

$\mathrm{evalf}\left(\=\right)$

$0.06781184623=0.06781184623$

[1] J. Bluemlein, D.J. Broadhurst, J.A.M. Vermaseren.  "The Multiple Zeta Value Data Mine", Comput.Phys.Commun. Vol. 181 (2010): p. 582-625.

The MultiZeta command was introduced in Maple 2018.