Hurwitz Zeta Function

Author

Eric W. Weisstein
September 19, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

(∑_(k=0)^∞)(1/(((a+k))^s))==(HurwitzZeta[s, a])

V6

Sum[(1/(((k+a))^s)),{k,0,∞}]

V7

Sum[(1/(((k+a))^s)),{k,0,∞}]

TraditionalForm[%]

(∑_(k=0)^∞)(1/((((((a+k))^2)))^(s/2)))==(SuperStar[ζ])(s,a)

Sum[(1/((((((k+a))^2)))^(s/2))),{k,0,∞}]

(1/(s-1))(∑_(n=0)^∞)(1/(n+1))(∑_(k=0)^n)(((-1))^k) (((a+k))^(1-s)) (Binomial[n, k])==(HurwitzZeta[s, a])

(1/(s-1))Sum[(1/(n+1))Sum[(((-1))^k)Binomial[n,k](((a+k))^(1-s)),{k,0,n}],{n,0,∞}]

(1/(s-1))Sum[((((-1))^k)/(n+1))Binomial[n,k](((a+k))^(1-s)),{n,0,∞},{k,0,n}]

(HurwitzZeta[s, (p/q)])==2 (((2 π q))^(s-1)) Γ(1-s)(∑_n^q)sin(((2 π n p)/q)+((π s)/2)) ζ(1-s,(n/q))

TimeConstrained[2Gamma[1-s](((2π q))^(s-1))Sum[Sin[((π s)/2)+((2π n p)/q)]HurwitzZeta[1-s,(n/q)],{n,q}],60]//Timing

l=Select[Sort/@RandomInteger[{1,5},{10,2}],CoprimeQ@@#&]

Subtract@@@Table[{HurwitzZeta[s,p/q],2Gamma[1-s](((2π q))^(s-1))Sum[Sin[((π s)/2)+((2π n p)/q)]HurwitzZeta[1-s,(n/q)],{n,q},Method->"Procedural"]}/.{s->RandomComplex[],p->l[[i,1]],q->l[[i,2]]},{i,5}]//Chop//Union

(((2 Γ(1-z)) )/(((2 π))^(2-z)))(sin(((π z)/2)) (∑_(n=1)^∞)((cos(2 π a n))/(n^(1-z)))+cos(((π z)/2)) (∑_(n=1)^∞)((sin(2 π a n))/(n^(1-z))))==(HurwitzZeta[z, a])

((2Gamma[1-z])/(((2π))^(1-z)))(Sin[((π z)/2)]Sum[((Cos[2π a n])/(n^(1-z))),{n,1,∞}]+Cos[((π z)/2)]Sum[((Sin[2π a n])/(n^(1-z))),{n,1,∞}])

FullSimplify[%]

Subtract@@@Table[{I (E^(-(1/2) I π z)) (((2 π))^(-1+z)) Gamma[1-z] (PolyLog[1-z,(E^(-2 I a π))]-(E^(I π z)) PolyLog[1-z,(E^(2 I a π))]),HurwitzZeta[z,a]}/.{z->RandomComplex[],a->RandomComplex[]},{100}]//Chop//Union

((2Gamma[1-z])/(((2π))^(1-z)))(Sum[Sin[((π z)/2)]((Cos[2π a n])/(n^(1-z)))+Cos[((π z)/2)]((Sin[2π a n])/(n^(1-z))),{n,1,∞}])

FullSimplify[%]

(∑_(k=0)^∞)(((k+(1/2)))^(-s))==((2^s)-1) ζ(s)

Sum[(((k+(1/2)))^(-s)),{k,0,∞}]

(2^s) (∑_(k=0)^∞)(((2 k+1))^(-s))==((2^s)-1) ζ(s)

(2^s)Sum[(((2k+1))^(-s)),{k,0,∞}]

(2^s) (ζ(s)-(∑_k^∞)(((2 k))^(-s)))==((2^s)-1) ζ(s)

(2^s)(Zeta[s]-Sum[(((2k))^(-s)),{k,∞}])

FullSimplify[%]

Integrals

ζ(s,a)==(1/(Γ(s)))(∫_0^∞)((t^(s-1))/((e^(a t)) (1-(e^(-t)))))dt/;Re[s]>1∧Re[a]>0

V6

(1/(Gamma[s]))Integrate[((t^(s-1))/(Exp[a t](1-Exp[-t]))),{t,0,∞}]

(1/(Gamma[s]))Integrate[((t^(s-1))/(Exp[a t](1-Exp[-t]))),{t,0,∞},Assumptions->Re[a]>0&&Re[s]>1]

V7

(1/(Gamma[s]))Integrate[((t^(s-1))/(Exp[a t](1-Exp[-t]))),{t,0,∞}]

(1/(Gamma[s]))Integrate[((t^(s-1))/(Exp[a t](1-Exp[-t]))),{t,0,∞},Assumptions->Re[a]>0&&Re[s]>1]

(HurwitzZeta[s, a])==((a^(1-s))/(s-1))+((a^(-s))/2)+2 (∫_0^∞)((((((a^2)+(y^2)))^(-(s/2))) sin(s (tan^(-1))((y/a))))/(exp(2 π y)-1))dy

(1/2)(a^(-s))+((a^(1-s))/(s-1))+2Integrate[((((a^2)+(y^2)))^(-s/2))(Sin[s ArcTan[(y/a)]])(1/(Exp[2π y]-1)),{y,0,∞}]//Timing

Special Values

Flatten[Outer[{#1,#2,Zeta[s,#1,IncludeSingularTerm->#2]}&,{0,1},{True,False}],1]//TableForm

Zeta[s,#]&/@{0,1}

HurwitzZeta[s,#]&/@{0,1}

(SuperStar[ζ])(s,0)==ζ(s)

Zeta[s,0]

HurwitzZeta[s,0]

(SuperStar[ζ])(s,1)==(HurwitzZeta[s, 1])==ζ(s)

Zeta[s,1]

HurwitzZeta[s,1]

ζ(-n,a)=-(((Subscript[B, n+1])(a))/(n+1))

Subtract@@@FunctionExpand[Table[{-BernoulliB[n+1,a]/(n+1),Zeta[-n,a]},{n,10}]]

FunctionExpand[HurwitzZeta[-n,a],n≥0&&n∈Integers]

FunctionExpand[Zeta[-n,a],n≥0&&n∈Integers]

(HurwitzZeta[0, a])==(1/2)-a

Zeta[0,a]//FunctionExpand

HurwitzZeta[0,a]//FunctionExpand

Derivative Identities

((∂ζ(s,a))/(∂a))==-s ζ(s+1,a)

V7

Derivative[0,1][Zeta][s,a]

Derivative[0,1][HurwitzZeta][s,a]

Derivative[1,0][Zeta](0,a)==log(Γ(a))-(1/2) log(2 π)

V5.2

Derivative[1,0][Zeta][0,a]//FunctionExpand//FullSimplify

V6

Derivative[1,0][Zeta][0,a]//FunctionExpand//InputForm

V7

Derivative[1,0][Zeta][0,a]//FunctionExpand

Derivative[1,0][HurwitzZeta][0,a]//FunctionExpand

FullSimplify[%,a>1]

FullSimplify[Subtract@@{-(1/2) Log[2 π]+Log[Gamma[a]],Log[-1+a]-(1/2) Log[2 π]+LogGamma[-1+a]},a>1]

Derivative[1,0][ζ](0,0)==-(1/2) ln(2 π)

V5.2

Derivative[1,0][Zeta][0,0]//FunctionExpand//FullSimplify

V6

Derivative[1,0][Zeta][0,0]//FunctionExpand//InputForm

V7

Derivative[1,0][Zeta][0,0]//FunctionExpand//InputForm

Derivative[1,0][HurwitzZeta][0,0]//FunctionExpand

Derivative[1,0][Zeta](1-2 k,(1/2))==-((Zeta'(1-2 k) ((2^(2 k-1))-1))/(2^(2 k-1)))+(((ln2) (-(Subscript[B, 2 k])))/((4^k) k))

Table[{Derivative[1,0][Zeta][-2k+1,1/2],-BernoulliB[2k]Log[2]/4^k/k -(2^(2k-1)-1)Derivative[1][Zeta][-2k+1]/(2^(2k-1))} ,{k,5}]//FullSimplify

N[Subtract@@@%]//Chop

Derivative[1,0][Zeta](1-2 k,(1/3))==-((3 log(3) (Subscript[B, 2 k]))/(4 k (9^k)))-(((Sqrt[3]) ((9^k)-1) π (Subscript[B, 2 k]))/(8 k (9^k)))-((((9^k)-3) Zeta'(1-2 k))/(2 (9^k)))-(((((-1))^k) (ψ^((2 k-1)))((1/3)))/(2 (Sqrt[3]) (((6 π))^(2 k-1))))

Table[{
Derivative[1,0][Zeta][1-2k,1/3],
((-(Sqrt[3])((9^k)-1)BernoulliB[2k]π)/(8k (9^k)))-((BernoulliB[2k]3Log[3])/(4k (9^k)))-(((((-1))^k)PolyGamma[2k-1,(1/3)])/(2(Sqrt[3])(((6π))^(2k-1))))-((((9^k)-3)Zeta'[1-2k])/(2 (9^k)))
},{k,1,5}]//FullSimplify

N[Subtract@@@%,10]//Chop

Derivative[1,0][Zeta](1-2 k,(2/3))==-((3 log(3) (Subscript[B, 2 k]))/(4 k (9^k)))+(((Sqrt[3]) ((9^k)-1) π (Subscript[B, 2 k]))/(8 k (9^k)))+(((((-1))^k) (ψ^((2 k-1)))((1/3)))/(2 (Sqrt[3]) (((6 π))^(2 k-1))))-((((9^k)-3) Zeta'(1-2 k))/(2 (9^k)))

Table[{
Derivative[1,0][Zeta][1-2k,2/3],
(((Sqrt[3])((9^k)-1)BernoulliB[2k]π)/(8k (9^k)))-((BernoulliB[2k]3Log[3])/(4k (9^k)))+(((((-1))^k)PolyGamma[2k-1,(1/3)])/(2(Sqrt[3])(((6π))^(2k-1))))-((((9^k)-3)Zeta'[1-2k])/(2 (9^k)))
},{k,1,5}]//FullSimplify

N[Subtract@@@%,10]//Chop

Derivative[1,0][Zeta](1-2 k,(1/4))==((((4^(k-1))-1) log(2) (Subscript[B, 2 k]))/(k (2^(4 k-1))))-((((4^k)-1) π (Subscript[B, 2 k]))/(k (4^(k+1))))-((((4^k)-2) Zeta'(1-2 k))/(2^(4 k)))-(((((-1))^k) (ψ^((2 k-1)))((1/4)))/(4 (((8 π))^(2 k-1))))

Table[{
Derivative[1,0][Zeta][1-2k,(1/4)],
-((((4^k)-1)BernoulliB[2k]π)/(k (4^(k+1))))+((((4^(k-1))-1)BernoulliB[2k]Log[2])/(k (2^(4k-1))))-(((((-1))^k)PolyGamma[2k-1,(1/4)])/(4(((8π))^(2k-1))))-((((4^k)-2)Zeta'[1-2k])/(2^(4k)))
},{k,1,5}]//FullSimplify

N[Subtract@@@%,10]//Chop

Derivative[1,0][Zeta](1-2 k,(3/4))==((((4^(k-1))-1) log(2) (Subscript[B, 2 k]))/(k (2^(4 k-1))))+((((4^k)-1) π (Subscript[B, 2 k]))/(k (4^(k+1))))-((((4^k)-2) Zeta'(1-2 k))/(2^(4 k)))+(((((-1))^k) (ψ^((2 k-1)))((1/4)))/(4 (((8 π))^(2 k-1))))

Table[{
Derivative[1,0][Zeta][1-2k,(3/4)],
((((4^k)-1)BernoulliB[2k]π)/(k (4^(k+1))))+((((4^(k-1))-1)BernoulliB[2k]Log[2])/(k (2^(4k-1))))+(((((-1))^k)PolyGamma[2k-1,(1/4)])/(4(((8π))^(2k-1))))-((((4^k)-2)Zeta'[1-2k])/(2^(4k)))
},{k,1,5}]//FullSimplify

N[Subtract@@@%,10]//Chop

Derivative[1,0][Zeta](1-2 k,(1/6))==((((3^(2 k-1))-1) log(2) (Subscript[B, 2 k]))/(4 k (6^(2 k-1))))+((((2^(2 k-1))-1) log(3) (Subscript[B, 2 k]))/(4 k (6^(2 k-1))))-((((9^k)-1) ((2^(2 k-1))+1) π (Subscript[B, 2 k]))/(8 k (Sqrt[3]) (6^(2 k-1))))+((((2^(2 k-1))-1) ((3^(2 k-1))-1) Zeta'(1-2 k))/(2 (6^(2 k-1))))-(((((-1))^k) ((2^(2 k-1))+1) (ψ^((2 k-1)))((1/3)))/(2 (Sqrt[3]) (((12 π))^(2 k-1))))

Table[{
Derivative[1,0][Zeta][1-2k,(1/6)],
-((((9^k)-1)((2^(2k-1))+1)BernoulliB[2k]π)/(8k(Sqrt[3])(6^(2k-1))))+((((3^(2k-1))-1)BernoulliB[2k]Log[2])/(4k (6^(2k-1))))+((((2^(2k-1))-1)BernoulliB[2k]Log[3])/(4k (6^(2k-1))))-(((((-1))^k)((2^(2k-1))+1)PolyGamma[2k-1,(1/3)])/(2(Sqrt[3])(((12π))^(2k-1))))+((((2^(2k-1))-1)((3^(2k-1))-1)Zeta'[1-2k])/(2 (6^(2k-1))))
},{k,1,5}]

N[Subtract@@@%,10]//Chop

Derivative[1,0][Zeta](1-2 k,(5/6))==((((3^(2 k-1))-1) log(2) (Subscript[B, 2 k]))/(4 k (6^(2 k-1))))+((((2^(2 k-1))-1) log(3) (Subscript[B, 2 k]))/(4 k (6^(2 k-1))))+((((9^k)-1) ((2^(2 k-1))+1) π (Subscript[B, 2 k]))/(8 k (Sqrt[3]) (6^(2 k-1))))+((((2^(2 k-1))-1) ((3^(2 k-1))-1) Zeta'(1-2 k))/(2 (6^(2 k-1))))+(((((-1))^k) ((2^(2 k-1))+1) (ψ^((2 k-1)))((1/3)))/(2 (Sqrt[3]) (((12 π))^(2 k-1))))

Table[{
Derivative[1,0][Zeta][1-2k,(5/6)],
((((9^k)-1)((2^(2k-1))+1)BernoulliB[2k]π)/(8k(Sqrt[3])(6^(2k-1))))+((((3^(2k-1))-1)BernoulliB[2k]Log[2])/(4k (6^(2k-1))))+((((2^(2k-1))-1)BernoulliB[2k]Log[3])/(4k (6^(2k-1))))+(((((-1))^k)((2^(2k-1))+1)PolyGamma[2k-1,(1/3)])/(2(Sqrt[3])(((12π))^(2k-1))))+((((2^(2k-1))-1)((3^(2k-1))-1)Zeta'[1-2k])/(2 (6^(2k-1))))
},{k,1,5}]

N[Subtract@@@%,10]//Chop

Plots

ListContourPlot

ListContourPlot[Table[N[Zeta[s,a]/.ComplexInfinity->0,20],{s,-2,2,5/100},{a,-5,5,5/100}],
ContourLines->False,ColorFunction->Hue,
FrameLabel->TraditionalForm/@{s,a},PlotLabel->TraditionalForm[Zeta[s,a]]]//Timing

DensityPlot[N[Zeta[s,a]/.ComplexInfinity->0,20],{a,-5,5},{s,-2,2},PlotPoints->200,
ColorFunction->Hue,Mesh->False,
FrameLabel->TraditionalForm/@{s,a},PlotLabel->TraditionalForm[Zeta[s,a]]]//Timing

Show[Block[{$DisplayFunction=Identity},
{
ContourPlot[N[Zeta[s,a]/.ComplexInfinity->0,20],{s,-2,2},{a,-5,5},PlotPoints->300,ContourLines->False,
Contours->30,ColorFunction->Hue,
FrameLabel->TraditionalForm/@{s,a},PlotLabel->TraditionalForm[Zeta[s,a]]],
ContourPlot[N[Zeta[s,a]/.ComplexInfinity->0,20],{s,-2,2},{a,-5,5},PlotPoints->300,Contours->{0},ContourShading->False]
}],PlotRange->All
]//Timing

Need to be Careful with Numerical Precision

ContourPlot[Zeta[s,a],{s,-10,0},{a,1/2,2},ContourLines->False,PlotPoints->200,ColorFunction->Hue]//Timing

ListContourPlot[Table[Zeta[s,a]/.ComplexInfinity->0,{s,-10,0,.1},{a,1/2,2,.1}],
ContourLines->False,ColorFunction->Hue]

ListContourPlot[Table[N[Zeta[s,a]/.ComplexInfinity->0,20],{s,-10,0,1/10},{a,1/2,2,1/10}],
ContourLines->False,ColorFunction->Hue]//Timing

Limits

SingularTerm

v4.2

Limit[Zeta[0,a,IncludeSingularTerm->True],a->0,Direction->1]

{
Zeta[#,0],
Zeta[#,0,IncludeSingularTerm->True],
Limit[Zeta[#,a],a->0,Direction->-1],
Limit[Zeta[#,a,IncludeSingularTerm->True],a->0,Direction->-1],
Limit[Zeta[#,a],a->0,Direction->1],
Limit[Zeta[#,a,IncludeSingularTerm->True],a->0,Direction->-1],
}&/@Range[0,10]

v5.0

{
Zeta[#,0],
Zeta[#,0,IncludeSingularTerm->True],
Limit[Zeta[#,a],a->0,Direction->-1],
Limit[Zeta[#,a],a->0,Direction->1]
}&/@Range[0,10]//TableForm

v5.1

{
Zeta[#,0],
Zeta[#,0,IncludeSingularTerm->True],
Limit[Zeta[#,a],a->0,Direction->-1],Limit[Zeta[#,a],a->0,Direction->1]
}&/@Range[0,10]//TableForm

(lim_(s->1)) (ζ(s,a)-(1/(s-1)))

v5.0

Limit[Zeta[s,a]-(1/(s-1)),s->1]

v5.1

Limit[Zeta[s,a]-(1/(s-1)),s->1]

FullSimplify[-D[Gamma[a],a]/Gamma[a]]

Limit[Zeta[s,a,IncludeSingularTerm->#]-(1/(s-1)),s->1]&/@{True,False}

v5.2

Limit[Zeta[s,a]-(1/(s-1)),s->1]

Limit[Zeta[s,a,IncludeSingularTerm->#]-(1/(s-1)),s->1]&/@{True,False}

v6

Limit[Zeta[s,a]-(1/(s-1)),s->1]

Limit[Zeta[s,a,IncludeSingularTerm->#]-(1/(s-1)),s->1]&/@{True,False}

V7

Limit[Zeta[s,a]-(1/(s-1)),s->1]

Limit[HurwitzZeta[s,a]-(1/(s-1)),s->1]