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MultiZeta

the multiple zeta function

 Calling Sequence MultiZeta(${m}_{1},{m}_{2},...,{m}_{n})$

Parameters

 ${m}_{1},{m}_{2},...,{m}_{n}$ - positive integers

Description

MultiZeta is an implementation of multiple zeta values, also known as the generalized Euler sums over

$\mathrm{MultiZeta}\left({m}_{i}$\mathrm{=}\left(i,1..n\right)\right)=\mathrm{%sum}\left(\mathrm{Multiply}\left(\mathrm{/}\left(1,\mathrm{^}\left({i}_{j},{m}_{j}\right)\right)$\mathrm{=}\left(j,1..n\right)\right),i\right)$

 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.

Examples

 For one argument, MultiZeta reduces to the Riemann Zeta function:
 > $\mathrm{%MultiZeta}\left(43\right)=\mathrm{MultiZeta}\left(43\right)$
 ${\mathrm{%MultiZeta}}{}\left({43}\right){=}{\mathrm{\zeta }}{}\left({43}\right)$ (1)

The more relevant special cases are computed automatically, such as that of two identical arguments, here using a more compact input syntax

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

and of two arguments summing to an odd number

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

All Multiple Zeta values of weight less than or equal to seven, can be written solely in terms of classical Zeta values:

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

The multiple Zeta values are a special case of the multiple polylogarithm:

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

The multiple zeta values obey a large number of identities, primarily the stuffle relation:

 > $\mathrm{MultiZeta}\left(7,9\right)\mathrm{MultiZeta}\left(6\right)$
 $\frac{{\mathrm{MultiZeta}}{}\left({7}{,}{9}\right){}{{\mathrm{\pi }}}^{{6}}}{{945}}$ (6)
 > $\mathrm{MultiZeta}\left(7,9,6\right)+\mathrm{MultiZeta}\left(7,6,9\right)+\mathrm{MultiZeta}\left(6,7,9\right)+\mathrm{MultiZeta}\left(13,9\right)+\mathrm{MultiZeta}\left(7,15\right)$
 ${\mathrm{MultiZeta}}{}\left({7}{,}{9}{,}{6}\right){+}{\mathrm{MultiZeta}}{}\left({7}{,}{6}{,}{9}\right){+}{\mathrm{MultiZeta}}{}\left({6}{,}{7}{,}{9}\right){+}{\mathrm{MultiZeta}}{}\left({13}{,}{9}\right){+}{\mathrm{MultiZeta}}{}\left({7}{,}{15}\right)$ (7)

Up to 5 digits,

 > $\mathrm{evalf}\left[5\right]\left(=\right)$
 ${0.0084952}{=}{0.0084952}$ (8)

and the duality

 > $\mathrm{MultiZeta}\left(2,3,4\right)$
 ${\mathrm{MultiZeta}}{}\left({2}{,}{3}{,}{4}\right)$ (9)
 > $\mathrm{MultiZeta}\left(2,1,1,2,1,2\right)$
 ${\mathrm{MultiZeta}}{}\left({2}{,}{1}{,}{1}{,}{2}{,}{1}{,}{2}\right)$ (10)
 > $\mathrm{evalf}\left(=\right)$
 ${0.06781184623}{=}{0.06781184623}$ (11)

References

 [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.

Compatibility

 • The MultiZeta command was introduced in Maple 2018.