Multivariate Zeta Function

Author

Eric W. Weisstein
April 18, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/MultivariateZetaFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MultivariateZetaFunction.html.

©2011 Wolfram Research, Inc. except for portions noted otherwise

Code

MultipleSum[expr_,its__List]:=Fold[sum,expr,Reverse[{its}]]

MZV[s__]:=Module[{n=Length[{s}]},
MultipleSum[Times@@MapIndexed[Sign[#1]^(a[#2[[1]]])/a[#2[[1]]]^Abs[#1]&,{s}],
Evaluate[Sequence@@({a[#],1,a[#-1]-1}&/@Range[n])]] /.a[0]->Infinity
]

zeta[l__]:=Module[{n=Length[{l}]/2},
2Pi^(4n)/(4n+2)!
]/;Union[Partition[{l},2,2,{1,1},{}]]==={{3,1}}

Special Cases

(Zeta2[-3, -1])=-(Subscript[a, a])(1,3)==-(2 (PolyLog[4, (1/2)])-((Zeta[4])/2)-(1/2) (Zeta[2]) (log^2)(2)+(((log^4)(2))/12))

Numeric

With[{n=200},Total[((((-1))^(#1+#2))/(((#1)^3)((#2)^1)))&@@@Select[Flatten[Outer[List,Range[n],Range[n]],1],Greater@@#&]]]//N

NSum[-(((-1)^k*(Log[4]-(-1)^k*PolyGamma[0,k/2]+(-1)^k*PolyGamma[0,(1+k)/2]))/(2*k^3)),{k,Infinity},Method->Fit,WorkingPrecision->500,NSumTerms->500,NSumExtraTerms->500]//Chop//Timing

eulerSumAA[1,3]:=Log[2]^4/12-Zeta[4]/2-Zeta[2]Log[2]^2/2+2PolyLog[4,1/2]

N[eulerSumAA[1,3],20]

MZV[-3,-1]

%/.{a[1]->i,a[2]->j}

Need to evaluate this in V6

Sum[((((-1))^(i+j))/((i^3) j)),{j,1,-1+i}]

because in V7/V8, it returns

Sum[((((-1))^(i+j))/((i^3) j)),{j,1,-1+i}]

Can'tbe done in V6, V7, or V8

FullSimplify[(((((-1))^i) (-((Log[4])/2)+(1/2) (((-1))^i) (PolyGamma[0,(i/2)]-PolyGamma[0,((1+i)/2)])))/(i^3)),i∈Integers&&i>0]

Sum[(((((-1))^i) (-Log[4]+(((-1))^i) (PolyGamma[0,(i/2)]-PolyGamma[0,((1+i)/2)])))/(2 (i^3))),{i,∞}]//Timing

NSum[(((((-1))^i) (-Log[4]+(((-1))^i) (PolyGamma[0,(i/2)]-PolyGamma[0,((1+i)/2)])))/(2 (i^3))),{i,∞}]

V8

MZV[-3,-1]

%/.sum->Sum

ζ(-2,-1)==(3/2) ζ(2) log(2)-((13 ζ(3))/8)

Numeric

(3/2)*Zeta[2]*Log[2] - (13*Zeta[3])/8//N

With[{n=100},Total[(((((-1))^(#1))(((-1))^(#2)))/(((#1)^2)((#2)^1)))&@@@Select[Flatten[Outer[List,Range[n],Range[n]],1],Greater@@#&]]]//N

V6

MZV[-2,-1]

FullSimplify[%/.Sum->sum]/.a$8043[n_]:>{k,j,i}[[n]]/.sum->Sum

N[%]

V8

MZV[-2,-1]

%/.sum->Sum

(∑_(k=1)^∞)(((((-1))^k) (-Log[4]+(((-1))^k) (PolyGamma[0,(k/2)]-PolyGamma[0,((1+k)/2)])))/(2 (k^2)))

ζ(-2,1)==((ζ(3))/8)

Numerical

With[{n=200},Total[((((-1))^(#1))/(((#1)^2)((#2)^1)))&@@@Select[Flatten[Outer[List,Range[n],Range[n]],1],Greater@@#&]]]//N

Zeta[3]/8.

v5.2

MZV[-2,1]

...wrong...

v6

MZV[-2,1]//FullSimplify

MZV[-2,1]

FullSimplify[%]

N[%]

V7

MZV[-2,1]

V8

MZV[-2,1]/.sum->Sum

ζ(2,-1)==ζ(3)-(3/2) ζ(2) log(2)

V6

FullSimplify[sum[((((-1))^(a[2]))/(((a[1])^2) a[2])),{a[2],1,-1+a[1]}]/.{a[1]->i,a[2]->j}/.sum->Sum]

V8

MZV[2,-1]

sum[((((-1))^(a[2]))/(((a[1])^2) a[2])),{a[2],1,-1+a[1]}]/.{a[1]->i,a[2]->j}/.sum->Sum

Sum[((-Log[4]+(((-1))^i) (PolyGamma[0,(i/2)]-PolyGamma[0,((1+i)/2)]))/(2 (i^2))),{i,∞}]

ζ(2,1)==ζ(3)

V5.2

Numeric

With[{n=5000},Total[(1/(((#1)^2)((#2)^1)))&@@@Select[Flatten[Outer[List,Range[n],Range[n]],1],Greater@@#&]]]//N

N[Zeta[3]]

MZV[2,1]

...wrong...

V6

MZV[2,1]

V8

MZV[2,1]

MZV[2,1]/.sum->Sum

ζ(2,1,1)==ζ(4)

MZV[2,1,1]

%/.Sum->sum/.a$15[n_]:>{k,j,i}[[n]]/.sum->Sum

N[%]

...wrong...

Sum[HarmonicNumber[-1 + j]/(j*k^2), {k, 1000}, {j, 1, -1 + k}]//N

Zeta[4]//N

ζ(2,1,2,1)

MZV[2,1,2,1]

%/.sum->Sum

Sum[(1/(((a[1])^2) a[2] ((a[3])^2) a[4])),{a[4],1,-1+a[3]}]

Sum[((HarmonicNumber[-1+a[3]])/(((a[1])^2) a[2] ((a[3])^2))),{a[3],1,-1+a[2]}]

Sum[((HarmonicNumber[-1+a[3]])/(((a[1])^2) a[2] ((a[3])^2))),{a[3],1,-1+a[2]}]

%/.Sum->sum/.a$2863[n_]:>{k,j,i}[[n]]/.sum->Sum

N[MZV[2,1,2,1]/.Infinity->10]

(Zeta2[3, 1])==((π^4)/360)

MZV[3,1]

(1/2)Integrate[-((Log[1-x])/(1-x))Log[x]^2,{x,0,1}]

(Subscript[s, a])(2,3)==ζ(3,2)+2ζ(3,-1,-1)

Exact expression

N[4PolyLog[5,(1/2)]-(1/30)((Log[2])^5)-(17/32)Zeta[5]-(11/720)(π^4)Log[2]+(7/4)Zeta[3]((Log[2])^2)+(1/18)(π^2)((Log[2])^3)-(1/8)(π^2)Zeta[3],10]

As an Euler sum

Bailey et al. 2006, p. 39; Bailey et al. 2006

EulerSumSA[m_,n_]:=Sum[((1/2)*(-1)^k(HarmonicNumber[(1/2)*(-1+k)]-HarmonicNumber[k/2])+Log[2])^m (k+1)^(-n),{k,Infinity}]

EulerSumSA[2,3]

N[%]

LLL

<<Utilities`Simplify`

TranscendentalRecognize[
N[4PolyLog[5,(1/2)]-(1/30)((Log[2])^5)-(17/32)Zeta[5]-(11/720)(π^4)Log[2]+(7/4)Zeta[3]((Log[2])^2)+(1/18)(π^2)((Log[2])^3)-(1/8)(π^2)Zeta[3],50],
{(π^5),(π^4)Log[2],(π^3)((Log[2])^2),(π^2)((Log[2])^3),π ((Log[2])^4),((Log[2])^5),(π^2)Zeta[3],π Log[2]Zeta[3],((Log[2])^2)Zeta[3],Zeta[5],PolyLog[5,(1/2)]}]

Numerical Check of identity

N[Sum[(-1)^(i+j)/(i*j*k^3),{k,15},{j,k-1},{i,k-1}]]

N[Sum[((((-1))^(i+j))/(i j (k^3))),{k,150},{j,k-1},{i,k-1}],10]//Timing

N[2*Sum[(-1)^(i+j)/(k^3*j*i),{k,15},{j,k-1},{i,j-1}]+Sum[1/(j^3*i^2),{j,15},{i,j-1}]]

sum=2MZV[3,-1,-1]+MZV[3,2]

N[sum/.Infinity->50,10]//Timing

N[sum/.Infinity->1000,10]//Timing

Explicit sums

Sum[((((-1))^(i+j))/(i j (k^3))),{i,k-1}]//FullSimplify

Sum[Evaluate[%],{j,k-1}]//FullSimplify

Sum[Evaluate[%],{k,∞}]

ζ(3,1,3,1)

MZV[3,1,3,1]

%/.Sum->sum/.a$12501[n_]:>{k,j,i}[[n]]/.sum->Sum

zeta[3,1,3,1]

N[%]

With[{n=1000},
NSum[(1/((i^3)j (k^3)l)),{l,n},{k,l+1,n},{j,k+1,n},{i,j+1,n}]
]//Timing

ζ(3,2)

MZV[3,2]

%/.Sum->sum/.a$3039[n_]:>{k,j,i}[[n]]/.sum->Sum