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MultiZeta

the multiple zeta function

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

MultiZeta(m1,m2,...,mn)

Parameters

m1,m2,...,mn

-

positive integers

Description

MultiZeta is an implementation of multiple zeta values, also known as the generalized Euler sums over i1>i2>...>i__n >0

MultiZetami$`=`i,1..n=%sumMultiply`/`1,`^`ij,mj$`=`j,1..n,i

 

  

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:

%MultiZeta43=MultiZeta43

%MultiZeta43=ζ43

(1)

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

%MultiZeta=MultiZeta27,27

%MultiZeta27,27=ζ2722ζ542

(2)

and of two arguments summing to an odd number

%MultiZeta=MultiZeta11,8;

%MultiZeta11,8=75583ζ192+9724π2ζ173+4433π4ζ1590+286π6ζ13315+121π8ζ119450+8π10ζ993555

(3)

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

%MultiZeta=MultiZeta2,1,4

%MultiZeta2,1,4=7π4ζ336011π2ζ512+61ζ78

(4)

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

%MultiPolylog=MultiPolylog2,3,4,5,1,1,1,1;

%MultiPolylog2,3,4,5,1,1,1,1=MultiZeta2,3,4,5

(5)

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

MultiZeta7,9MultiZeta6

MultiZeta7,9π6945

(6)

MultiZeta7,9,6+MultiZeta7,6,9+MultiZeta6,7,9+MultiZeta13,9+MultiZeta7,15

MultiZeta7,9,6+MultiZeta7,6,9+MultiZeta6,7,9+MultiZeta13,9+MultiZeta7,15

(7)

Up to 5 digits,

evalf5=

0.0084952=0.0084952

(8)

and the duality

MultiZeta2,3,4

MultiZeta2,3,4

(9)

MultiZeta2,1,1,2,1,2

MultiZeta2,1,1,2,1,2

(10)

evalf=

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.

See Also

GeneralizedPolylog

MultiPolylog

Zeta