Riemann Zeta Function

Author

Eric W. Weisstein
May 17, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Plots

Real Axis

V5

Show[GraphicsArray[{
Plot[Zeta[x],{x,-14.5,1},
PlotRange->{-.1,.1},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,Zeta[x]},
DisplayFunction->Identity],
PlotDiscontinuous[Zeta[x],{x,-10,1,10},
VerticalAsymptotes->{1},
DisplayFunction->Identity,
AxesLabel->TraditionalForm/@{x,Zeta[x]}]
},GraphicsSpacing->-.07
]]

V6

With[{opts=Sequence@@{PlotStyle->Red,AxesLabel->TraditionalForm/@{x,Zeta[x]},LabelStyle->Directive[Black,FontFamily->"Times",12]}},
GraphicsRow[{
Plot[Zeta[x],{x,-15,1},PlotRange->{-.1,.1},opts],
Plot[Zeta[x],{x,-10,10},opts,Epilog->{Dashing[Small],Line[{{1,-10},{1,10}}]}]
},ImageSize->500]]

Near origin

ComplexPlot3D[Zeta[z],{z,-(2I+2),3I+3},Mesh->20]

Plot3D[Abs[Zeta[x+I y]],{x,-1,3},{y,-3/2,3/2},PlotPoints->20,Mesh->50,MeshFunctions->{#3&},PlotRange->{All,All,{0,3}},BoxRatios->Automatic]

Plot3D[Re[Zeta[x+I y]],{x,-1/2,3},{y,-3/2,3/2},PlotPoints->100,Mesh->50,MeshFunctions->{#3&},PlotRange->{All,All,{-1,3}},BoxRatios->Automatic]

Plot3D[Im[Zeta[x+I y]],{x,-1/2,3},{y,-3/2,3/2},PlotPoints->20,Mesh->50,MeshFunctions->{#3&},PlotRange->{All,All,{-3,3}},BoxRatios->Automatic]

Complex Plane

V5 ReIm

ComplexPlot3D[Zeta[z],{z,-15,15},PlotPoints->150]//Timing

V5 Complex Plane Contour Plot

ComplexContourPlot[Zeta[z],{z,-15,15}]//Timing

n=50;Show[GraphicsArray[ContourPlot[#[Zeta[x+I y]],{x,-n,n},{y,-n,n},PlotPoints->200,
ContourLines->False,
DisplayFunction->Identity,ColorFunction->(Hue[#]&),FrameLabel->TraditionalForm/@{x,y,#[Zeta[x+I y]],""}]&/@{Re,Im}],GraphicsSpacing->-.05]

Plot3D[Abs[Zeta[x+I y]],{x,0,1},{y,1,100},PlotPoints->150,Mesh->False,PlotRange->All,AxesLabel->TraditionalForm/@{x,y,""},PlotLabel->TraditionalForm[Abs[Zeta[x+I y]]]]

V6

Plot3D[Abs[Zeta[x+I y]],{x,0,1},{y,1,100},Mesh->False,PlotRange->All,PlotPoints->{10,50},MaxRecursion->4,AxesLabel->{x,y,""},PlotLabel->Abs[Zeta[x+I y]]]

Critical Line

ListPlot[{Re[#],Im[#]}&/@Table[Zeta[1/2+y I],{y,0,35,.05}],
PlotJoined->True,PlotRange->All,AspectRatio->Automatic,AxesLabel->TraditionalForm/@{Re[Zeta[1/2+y I]],Im[Zeta[1/2+y I]]},PlotStyle->Red]

Pole

Residue[Zeta[z],{z,1}]

Renormalization

Zeta[-1]

Contour

Integrate[((u^(z-1))/(Exp[-u]-1)),{u,-∞-I ϵ,-I ϵ}]

D[ρ Exp[I θ],θ]

Integrate[((((ρ Exp[I θ]))^(z-1))/(Exp[-ρ Exp[I θ]]-1))I Exp[I θ]ρ,{θ,-π/2,π/2}]

Integrate[((u^(z-1))/(Exp[-u]-1)),{u,-∞+I ϵ,I ϵ}]

zeta[z_]:=With[{ϵ=10^-5,ρ=10^-5},
Plus@@{NIntegrate[((u^(z-1))/(Exp[-u]-1)),{u,-∞-I ϵ,-I ϵ}],
NIntegrate[((((ρ Exp[I θ]))^(z-1))/(Exp[-ρ Exp[I θ]]-1))I Exp[I θ]ρ,{θ,-π/2,π/2}],NIntegrate[((u^(z-1))/(Exp[-u]-1)),{u,I ϵ,-∞+I ϵ}]
}Gamma[1-z]/(2Pi I)
]

Through[{zeta,Zeta}[.2]]

Plot[{Zeta[x],zeta[x]},{x,-1.5,1.5},PlotStyle->{Red,Blue}]//Timing

Connection with Hurwitz

Zeta[s,1]

TableForm[Flatten[Outer[{#1,#2,Zeta[s,#1,IncludeSingularTerm->#2]}&,{0,1},{True,False}],1],TableHeadings->{None,{"a","IncludeSingularTerm","Result"}}]

Powers of 10 for Positive Even

Enumeration

Table[Digits[Denominator[Zeta[10^n]]],{n,0,6}]//Timing

Series

Series[Zeta[n],{n,∞,2}]

FunctionExpand[Zeta[n],n∈Integers&&n>0&&Mod[n,2]==0]

Table[1/(2^(z-1)(-1)^(1+z/2)BernoulliB[z]/z!),{z,2,10,2}]

expr=Log[10,(-1)^(-1-z/2)*2^(1-z)*z!];

ser=Series[expr,{z,∞,2}]

N[Normal[ser]/. z->10^5,10]

N[expr/.z->10^5,10]

...the series is wrong...

expr=Simplify[Normal[Series[Log[10,1/(2^(z-1)(-1)^(1+z/2)BernoulliB[z]/z!)],{z,∞,1}]]/.z->10^n,n∈Integers&&n>0]

Table[1+Floor[Re[expr]],{n,4}]

Primes

Position[Table[Denominator[Zeta[n]],{n,2,10^4,2}],_?PrimeQ]//Flatten//Timing

Position[Table[Denominator[Zeta[n]],{n,2,10^5,2}],_?PrimeQ]//Flatten//Timing

Integrals

(1/(Gamma[x]))(∫_0^∞)((u^(x-1))/(exp(u)-1))du==(Zeta[x])

(1/(Gamma[x]))Integrate[((u^(x-1))/(Exp[u]-1)),{u,0,∞}]

FullSimplify[%,Re[x]>1]

FullSimplify[((Exp[-u](u^(n-1)))/(1-Exp[-u]))]

Sum[Exp[-k u](u^(n-1)),{k,∞}]

(∫_0^∞)frac((1/t)) (t^(s-1))dt==-((Zeta[s])/s)

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

With[{s=.2},Plot[FractionalPart[(1/t)](t^(s-1)),{t,0,3},PlotStyle->Red,PlotPoints->50]]

With[{s=.2},Plot[FractionalPart[(1/t)](t^(s-1)),{t,0,.5},PlotStyle->Red,PlotPoints->50]]

With[{s=.2},{NIntegrate[FractionalPart[(1/t)](t^(s-1)),{t,0,.1,.2,1,2,∞},MaxRecursion->30,Method->{"GlobalAdaptive",MaxNumberOfErrorIncreases->10^4}],-((Zeta[s])/s)}]//Timing

With[{s=.2},NIntegrate[FractionalPart[(1/t)](t^(s-1)),{t,0,1,∞}]]

Integral Product

zeta[n_]:=Module[{a=Array[x,n]},
Integrate[1/(1-Times@@a),Sequence@@({#,0,1}&/@a)]
]

Table[zeta[n],{n,2,6}]

Table[Zeta[n],{n,2,6}]

Products

(∏_n^∞)(1/(1-(1/((((Prime[n])))^s))))==(Zeta[s])

V6

Product[(((1-(1/((Prime[n])^s))))^(-1)),{n,∞}]

Sums

(∑_k^∞)(1/(k^n))==(Zeta[n])

Sum[(1/(k^n)),{k,∞}]

(∑_(n=1)^∞)(1/(n^s))+(∑_(n=1)^∞)((((-1))^n)/(n^s))

V6

Re(s)>1

Sum[((((-1))^n)/(n^s)),{n,∞}]+Sum[(1/(n^s)),{n,∞}]//FullSimplify

2Sum[(1/(n^s)),{n,2,∞,2}]

2Sum[(1/(((2k))^s)),{k,∞}]

Simplify[Sum[(-1)^(n-1)/n^s,{n,∞}]/(1-2^(1-s))]

V7

Sum[((((-1))^n)/(n^s)),{n,∞}]+Sum[(1/(n^s)),{n,∞}]//FullSimplify

{SumConvergence[((((-1))^n)/(n^s)),n],SumConvergence[(1/(n^s)),n]}

(1/(1-(2^(1-s))))(∑_n^∞)((((-1))^(n-1))/(n^s))==(Zeta[s])

(1/(1-(2^(1-s))))Sum[((((-1))^(n-1))/(n^s)),{n,∞}]//Simplify

(∑_(n=0)^∞)(1/(2^(n+1)))(∑_(k=0)^n)(((((-1))^k) (Binomial[n, k]))/(((k+1))^z))==(Zeta[s])

Knopp-Sondow Globally Convergent Sum

ζ(z)==(1/(1-(2^(1-z))))(∑_(n=0)^∞)(1/(2^(n+1))) (∑_(k=0)^n)(((-1))^k) (({{n}, {k}})) (((k+1))^(-z))

v4.2

zeta[z_]:=1/(1-2^(1-z))*Sum[1/2^(n+1)*Sum[((-1)^k*Binomial[n,k])/(k+1)^z,{k,0,n}],{n,0,Infinity}]

FullSimplify[{zeta[2],zeta[3],zeta[4]}]

N[%]

v5.0

Sum[((-1)^k*Binomial[n,k])/(k+1)^z,{k,0,n}]

With[{z=2},Sum[((-1)^k*Binomial[n,k])/(k+1)^z,{k,0,n}]]

With[{z=4},Sum[((-1)^k*Binomial[n,k])/(k+1)^z,{k,0,n}]]

zeta[z_]:=1/(1-2^(1-z))*Sum[1/2^(n+1)*Sum[((-1)^k*Binomial[n,k])/(k+1)^z,{k,0,n}],{n,0,Infinity}]

zeta[s]

FullSimplify[{zeta[2],zeta[3],zeta[4]}]

N[%]

Table

Table[Zeta[n],{n,2,4}]

N[%]

v6.0

zeta[z_]:=1/(1-2^(1-z))*Sum[1/2^(n+1)*Sum[((-1)^k*Binomial[n,k])/(k+1)^z,{k,0,n}],{n,0,Infinity}]

zeta[s]

zeta/@Range[2,4]

FullSimplify[First[%]]

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

Hasse Globally Convergent Sum

zeta[s_]:=(1/(s-1))Sum[(1/(n+1))Sum[(((-1))^k)Binomial[n,k](((k+1))^(1-s)),{k,0,n}],{n,0,∞}]

V6.0

zeta/@Range[2,5]

FullSimplify[(1/(1+n))(((6 (EulerGamma^2)+(π^2))/(12 (1+n)))+(1/(2+2 n))(2 EulerGamma PolyGamma[0,2+n]+((PolyGamma[0,2+n])^2)-PolyGamma[1,2+n])),n∈Integers&&n>1]

Sum[%,{n,0,∞}]

(1/3)NSum[(((π^2)+6 ((HarmonicNumber[1+n])^2)-6 PolyGamma[1,2+n])/(12 (((1+n))^2))),{n,0,∞}]

Zeta[4.]

V6.1

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

Print[{#,zeta[#]//Timing}]&/@Range[2,4]

Sum[(((π^2)+6 ((HarmonicNumber[1+n])^2)-6 PolyGamma[1,2+n])/(12 (((1+n))^2))),{n,0,∞}]

(∑_(n=0)^∞)(((((-1))^n) (((s-1))^n) (StieltjesGamma[n]))/(n!))+(1/(s-1))==(Zeta[s])

(1/(s-1))+Sum[((((-1))^n)/(n!))StieltjesGamma[n](((s-1))^n),{n,0,∞}]

(∑_n^∞)((LiouvilleLambda[n])/(n^s))==((Zeta[2 s])/(Zeta[s]))

Sum[((LiouvilleLambda[n])/(n^s)),{n,∞}]

(∑_n^∞)((2^(PrimeNu[n]))/(n^s))==(((Zeta[s])^2)/(Zeta[2 s]))

Sum[((2^(PrimeNu[n]))/(n^s)),{n,∞}]//Timing

(1/2) (∑_(k=1)^∞)((Subscript[H, k])/(k^2))==ζ(3)

(1/2)Sum[((HarmonicNumber[k])/(k^2)),{k,∞}]

(lim_(x->∞)) (((∑_(k=1)^x)(cot^n)((k/(2 x+1))))/(((2 x+1))^n))==ζ(n)

Stark Identities

s[n_,x_]:=1/(2x+1)^n  Sum[Cot[k/(2x+1)]^n,{k,x}]

V6.0

sn[n_,x_]:=1/(2x+1)^n NSum[Cot[k/(2x+1)]^n,{k,x}]

Table[{sn[n,10000],Zeta[n]//N},{n,3,7,2}]

Limit[(1/(((2x+1))^n))Sum[((Cot[(k/(2x+1))])^n),{k,x}],x->∞]

Table[Limit[(1/(((2x+1))^n))Sum[((Cot[(k/(2x+1))])^n),{k,x}],x->∞],{n,2,3}]

(∑_(n=1)^∞)((μ(n))/(n^s))==(1/(ζ(s)))

Sum[((MoebiusMu[n])/(n^s)),{n,∞}]

-(∑_k^∞)((log(k))/(k^s))==ζ'(s)

-Sum[((Log[k])/(k^s)),{k,∞}]

-(∑_(k=2)^∞)((log(k))/(k^s))==ζ'(s)

-Sum[((Log[k])/(k^s)),{k,2,∞}]

(∑_k^∞)(((((-1))^k) (((s-1))^(k-1)) (StieltjesGamma[k]))/(k!))-(1/(((s-1))^2))==ζ'(s)

V7

sum=-(1/(((s-1))^2))+Sum[(1/(k!))StieltjesGamma[k](((-1))^k)(((s-1))^(k-1)),{k,∞}]

...wrong...

SumConvergence[(1/(k!))StieltjesGamma[k](((-1))^k)(((s-1))^(k-1)),k]

...wrong..

FullSimplify[sum,s>1]

sum/.s->4.

Zeta'[4.]

s1=-(1/(((s-1))^2))+Sum[(1/((k-1)!))StieltjesGamma[k](((-1))^k)(((s-1))^(k-1)),{k,5}]

s2=Series[Zeta'[s],{s,1,4}]

s1-s2

nodd, n ≡ 3 (mod 4)

(((2^(n-1))(π^n))/((n+1)!))Sum[(((-1))^(k-1))Binomial[n+1,2k]BernoulliB[n+1-2k]BernoulliB[2k],{k,0,(n+1)/2}]-2Sum[(1/((k^n)(Exp[2π k]-1))),{k,∞}]

Subtract@@@Table[{Zeta[n],(((2^(n-1))(π^n))/((n+1)!))Sum[(((-1))^(k-1))Binomial[n+1,2k]BernoulliB[n+1-2k]BernoulliB[2k],{k,0,(n+1)/2}]-2Sum[(1/((k^n)(Exp[2π k]-1))),{k,∞}]},{n,3,50,4}]//N[#,20]&

nodd, n ≡ 1 (mod 4)

((((2π))^n)/((n+1)!(n-1)))Sum[(((-1))^k)(n+1-4k)Binomial[n+1,2k]BernoulliB[n+1-2k]BernoulliB[2k],{k,0,(n+1)/4}]-2Sum[((Exp[2π k](1+((4π k)/(n-1)))-1)/((k^n)(((Exp[2π k]-1))^2))),{k,∞}]

Table[{Zeta[n],((((2π))^n)/((n+1)!(n-1)))Sum[(((-1))^k)(n+1-4k)Binomial[n+1,2k]BernoulliB[n+1-2k]BernoulliB[2k],{k,0,(n+1)/4}]-2Sum[((Exp[2π k](1+((4π k)/(n-1)))-1)/((k^n)(((Exp[2π k]-1))^2))),{k,∞}]},{n,5,20,4}]

%//N

(∑_(k=0)^n)(((((-1))^k) (({{n}, {k}})))/(((k+1))^4))

Sum[((-1)^k*Binomial[n, k])/(k + 1)^4, {k, 0, n}]

FullSimplify[%]

3 (∑_(k=1)^∞)(1/((k^2) (({{2 k}, {k}}))))==ζ(2)

{3Sum[(1/((k^2)Binomial[2k,k])),{k,∞}],Zeta[2]}

(5/2) (∑_(k=1)^∞)((((-1))^(k-1))/((k^3) (({{2 k}, {k}}))))==ζ(3)

{5/2Sum[((((-1))^(k-1))/((k^3)Binomial[2k,k])),{k,∞}],Zeta[3]}

(36/17) (∑_(k=1)^∞)(1/((k^4) (({{2 k}, {k}}))))==ζ(4)

v5.2

36/17Sum[(1/((k^4)Binomial[2k,k])),{k,∞}]

v6

{36/17Sum[(1/((k^4)Binomial[2k,k])),{k,∞}],Zeta[4]}

(∑_(n=2)^∞)(ζ(n)-1)==1

Sum[Zeta[n]-1,{n,2,∞}]

-(( 16 )/11)(∑_(n=1)^∞)(((2 (((-1))^n)+1) (Subscript[H, n]))/(n^4))==ζ(5)

-(16/11)Sum[(((2(((-1))^n)+1)HarmonicNumber[n])/(n^4)),{n,∞}]//FullSimplify

-(16/11)NSum[(((2(((-1))^n)+1)HarmonicNumber[n])/(n^4)),{n,∞}]

Zeta[5.]

(∑_(n=2, 4, …)^∞)(ζ(n)-1)==(3/4)

Sum[Zeta[n]-1,{n,2,∞,2}]

(∑_(n=3, 5, …)^∞)(ζ(n)-1)==(1/4)

Sum[Zeta[n]-1,{n,3,∞,2}]

(∑_(n=2)^∞)(((-1))^n)(ζ(n)-1)==(1/2)

Sum[(((-1))^n)(Zeta[n]-1),{n,2,∞}]

(∑_(n=1)^∞)(ζ(k n)-1)

Sum[Zeta[2k]-1,{k,∞}]//FullSimplify

Sum[Zeta[3k]-1,{k,∞}]//FullSimplify

Sum[Zeta[4k]-1,{k,∞}]

FullSimplify[%]

Sum[Zeta[5k]-1,{k,∞}]//FullSimplify

Sum[Zeta[6k]-1,{k,∞}]//FullSimplify

(∑_(k=2)^∞)((ζ(k)-1)/k)==1-ℽ

Sum[((Zeta[k]-1)/k),{k,2,∞}]

(∑_(k=2)^∞)((ζ(k) (((-1))^k))/k)==ℽ

(∑_(k=2)^∞)(((((-n))^k) (Zeta[k]))/k)==ℽ n+log((Gamma[n+1]))

V6

Sum[(((((-n))^k)Zeta[k])/k),{k,2,∞}]

FullSimplify[%,-1<n≤1]

V7

Sum[(((((-n))^k)Zeta[k])/k),{k,2,∞}]

(∑_(n=1)^∞)((ζ(2 n))/(n (2 n+1) (2^(2 n))))==log(π)-1

Sum[((Zeta[2n])/(n(2n+1)(2^(2n)))),{n,∞}]

(∑_(n=1)^∞)((ζ(2 n))/(n (2 n+1)))==log(2 π)-1

Sum[((Zeta[2n])/(n(2n+1))),{n,∞}]//Simplify

(∑_(k=1)^∞)((ζ(2 k,z))/(k (2 k+1) (2^(2 k))))==-2 z+(2 z-1) log(z-(1/2))-2 log((Gamma[z]))+log(2 π)+1

Sum[((Zeta[2k,z])/(k(2k+1)(2^(2k)))),{k,∞}]//FullSimplify

(∑_(k=2)^(n-2))((ζ(k) ζ(n-k))/(2^k))

B. Cloitre, Dec 11–13, 2005

Sum[Zeta[k](x^(k-1)),{k,2,∞}]

Sum[((Zeta[k]Zeta[n-k])/(2^k)),{k,2,n-2}]

Table[Sum[((Zeta[k]Zeta[n-k])/(2^k)),{k,2,n-2}],{n,3,10}]//TableForm//TraditionalForm

Table[Sum[((Zeta[k]Zeta[n-k])/(2^k)),{k,2,n-2}],{n,3,20}]//N

(∑_(k=1)^∞)(1/(((k (k+1)))^(2m+1)))== -2 (∑_(k=0)^m)(Binomial[4 m-2k+1, 2 m]) (InactiveHead[Zeta])(2 k)

Sum[(1/(((k(k+1)))^(2m+1))),{k,∞}]

m==1

sum=Sum[(1/(((k(k+1)))^3)),{k,∞}]

((CoefficientList[sum,(π^2)])/(Zeta[2Range[0,1]]))/.π->1

%.Zeta[2Range[0,1]]

m==2

sum=Sum[(1/(((k(k+1)))^5)),{k,∞}]//Expand

((CoefficientList[sum,(π^2)])/(Zeta[2Range[0,2]]))/.π->1

m==3

sum=Sum[(1/(((k(k+1)))^7)),{k,∞}]

((CoefficientList[sum,(π^2)])/(Zeta[2Range[0,3]]))/.π->1

m==4

sum=Sum[(1/(((k(k+1)))^9)),{k,∞}]

((CoefficientList[sum,(π^2)])/(Zeta[2Range[0,4]]))/.π->1

m==5

sum=Sum[(1/(((k(k+1)))^11)),{k,∞}]

((CoefficientList[sum,(π^2)])/(Zeta[2Range[0,5]]))/.π->1

m==6

sum=Sum[(1/(((k(k+1)))^13)),{k,∞}]

((CoefficientList[sum,(π^2)])/(Zeta[2Range[0,6]]))/.π->1

Initial term

2Table[Binomial[4n+1,2n],{n,6}]

FindSequenceFunction[{20,252,3432,48620,705432,10400600},n]

2nd Term

2Table[Binomial[4n-1,2n],{n,6}]

3rd term

seq={168,2574,38896,587860}

2Table[Binomial[4n-3,2n],{n,6}]

Last term

Table[4n+2,{n,5}]

Binomial[4n+2,4n+1]

Closed form

Table[-2Sum[Inactive[Zeta][2k]Binomial[4m-2k+1,2m],{k,0,m}],{m,6}]//Expand//Column

Table[Sum[(1/(((k(k+1)))^(2m+1))),{k,∞}]

Subtract@@@Table[{Sum[(1/(((k(k+1)))^(2m+1))),{k,∞}],-2Sum[Zeta[2k]Binomial[4m-2k+1,2m],{k,0,m}]},{m,6}]//Expand//Timing

With[{m=7},-2Sum[Zeta[2k]Binomial[4m-2k+1,2m],{k,0,m}]]

Expand[%-With[{m=7},Sum[(1/(((k(k+1)))^(2m+1))),{k,∞}]]]//Timing

Integer Arguments

Positive Even

Table[Zeta[n],{n,2,10,2}]

Table[2^(n-1)Abs[BernoulliB[n]]Pi^n/n!,{n,2,10,2}]

Table[2^(n-1)(-1)^(1+n/2)BernoulliB[n]Pi^n/n!,{n,2,10,2}]

FunctionExpand[Zeta[n],n∈Integers&&n>0&&Mod[n,2]==0]

Negative Odd

Table[Zeta[n],{n,-1,-10,-2}]

Table[-BernoulliB[-n+1]/(-n+1),{n,-1,-10,-2}]

FunctionExpand[Zeta[n],n∈Integers&&n<0&&Mod[n,2]==1]

Apéry-Type Sums

Sum[(1/(j^n)),{j,k-1}]

Even n

Sum[(1/((k^2)-(x^2))),{k,∞}]//Simplify

Sum[Zeta[2n+2](x^(2n)),{n,0,∞}]

Simplify[%,x>0]

Product[((1-((4(x^2))/(m^2)))/(1-((x^2)/(m^2)))),{m,k-1}]//FullSimplify

3Sum[(1/((k^2)Binomial[2k,k](1-((x^2)/(k^2)))))((Cos[π x] Gamma[k-2 x] Gamma[k+2 x])/(Gamma[k-x] Gamma[k+x])),{k,∞}]

Series[-((3 HypergeometricPFQ[{1,2,1-2 x,1+2 x},{(3/2),2-x,2+x},(1/4)])/(2 (-1+(x^2)))),{x,0,4}]

ζ(5)

Zeta[5]//N

2NSum[((((-1))^(k+1))/((k^5)Binomial[2k,k])),{k,∞}]-(5/2)NSum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,2],{k,∞}]

2Sum[((((-1))^(k+1))/((k^5)Binomial[2k,k])),{k,∞}]-(5/2)Sum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,2],{k,∞}]//Timing

ζ(7)

Zeta[7]//N

(5/2)NSum[((((-1))^(k+1))/((k^7)Binomial[2k,k])),{k,∞}]+(25/2)NSum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,4],{k,∞}]

(5/2)Sum[((((-1))^(k+1))/((k^7)Binomial[2k,k])),{k,∞}]+(25/2)Sum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,4],{k,∞}]//Timing

ζ(9)

Zeta[9]//N

(9/4)NSum[((((-1))^(k+1))/((k^9)Binomial[2k,k])),{k,∞}]-(5/4)NSum[((((-1))^(k+1))/((k^7)Binomial[2k,k]))HarmonicNumber[-1+k,2],{k,∞}]+5NSum[((((-1))^(k+1))/((k^5)Binomial[2k,k]))HarmonicNumber[-1+k,4],{k,∞}]+(45/4)NSum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,6],{k,∞}]-(25/4)NSum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,4]HarmonicNumber[-1+k,2],{k,∞}]

(9/4)Sum[((((-1))^(k+1))/((k^9)Binomial[2k,k])),{k,∞}]-(5/4)Sum[((((-1))^(k+1))/((k^7)Binomial[2k,k]))HarmonicNumber[-1+k,2],{k,∞}]+5Sum[((((-1))^(k+1))/((k^5)Binomial[2k,k]))HarmonicNumber[-1+k,4],{k,∞}]+(45/4)Sum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,6],{k,∞}]-(25/4)Sum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,4]HarmonicNumber[-1+k,2],{k,∞}]//Timing

ζ(11)

Zeta[11]//N

(5/2)NSum[((((-1))^(k+1))/((k^11)Binomial[2k,k])),{k,∞}]+(25/2)NSum[((((-1))^(k+1))/((k^7)Binomial[2k,k]))HarmonicNumber[-1+k,4],{k,∞}]-(75/4)NSum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,8],{k,∞}]+(125/4)NSum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))((HarmonicNumber[-1+k,4])^2),{k,∞}]

(5/2)Sum[((((-1))^(k+1))/((k^11)Binomial[2k,k])),{k,∞}]+(25/2)Sum[((((-1))^(k+1))/((k^7)Binomial[2k,k]))HarmonicNumber[-1+k,4],{k,∞}]-(75/4)Sum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))HarmonicNumber[-1+k,8],{k,∞}]+(125/4)Sum[((((-1))^(k+1))/((k^3)Binomial[2k,k]))((HarmonicNumber[-1+k,4])^2),{k,∞}]//Timing

Sums with No Known Closed Forms

(∑_(n=2)^∞)((ζ(n))/(n!))

Converges

Numeric

NSum[Zeta[n]/n!,{n,2,Infinity},WorkingPrecision->50]

v5

Sum[Zeta[n]/n!,{n,2,∞}]

Sum[1/n!/Gamma[n]u^(n-1)/(E^u-1),{n,2,∞}]

FullSimplify[%]

Integrate[((-u+(Sqrt[u]) BesselI[1,2 (Sqrt[u])])/((-1+(E^u)) u)),{u,0,∞}]

NIntegrate[((-u+(Sqrt[u]) BesselI[1,2 (Sqrt[u])])/((-1+(E^u)) u)),{u,0,10,∞},WorkingPrecision->120,MaxRecursion->20]//Timing

Integrate[((-1+Hypergeometric0F1Regularized[2,u])/(-1+(E^u))),{u,0,∞}]

NIntegrate[((-1+Hypergeometric0F1Regularized[2,u])/(-1+(E^u))),{u,0,10,∞},WorkingPrecision->120,MaxRecursion->20]//Timing

v6

Sum[Zeta[n]/n!,{n,2,∞}]

Integrate[((-u+(Sqrt[u]) BesselI[1,2 (Sqrt[u])])/((-1+(E^u)) u)),{u,0,∞}]

Integrate[((-1+Hypergeometric0F1Regularized[2,u])/(-1+(E^u))),{u,0,∞}]

NIntegrate[((-1+Hypergeometric0F1Regularized[2,u])/(-1+(E^u))),{u,0,10,∞},WorkingPrecision->120,MaxRecursion->20]//Timing

NIntegrate[((-u+(Sqrt[u]) BesselI[1,2 (Sqrt[u])])/((-1+(E^u)) u)),{u,0,10,∞},WorkingPrecision->120,MaxRecursion->20]//Timing

(∑_(n=1)^∞)((ζ(2 n))/(n!))

Numeric

Sum[x^(-2n)/n!,{x,Infinity}]

NSum[E^(1/x^2)-1,{x,Infinity},
Method->Integrate,
NSumTerms->50,NSumExtraTerms->20,WorkingPrecision->35]

v4.2

Wrong

Sum[E^(1/n^2)-1,{n,Infinity}]

v5.0

Sum[E^(1/n^2)-1,{n,Infinity}]

NSum[E^(1/x^2)-1,{x,1000},WorkingPrecision->110]//Timing

Series[E^(1/x^2)-1,{x,Infinity,20}]

NSum[((Zeta[2 n])/(n!)),{n,∞}]

Sum[1/n!/Gamma[2n]u^(2n-1)/(E^u-1),{n,∞}]

Integrate[((u HypergeometricPFQ[{},{(3/2),2},((u^2)/4)])/(-1+(E^u))),{u,0,Infinity}]

NIntegrate[((u HypergeometricPFQ[{},{(3/2),2},((u^2)/4)])/(-1+(E^u))),{u,0,Infinity}]

NIntegrate[((u HypergeometricPFQ[{},{(3/2),2},((u^2)/4)])/(-1+(E^u))),{u,0,Infinity},WorkingPrecision->110]//Timing

v6

Sum[Zeta[2n]/n!,{n,Infinity}]

Sum[E^(1/n^2)-1,{n,Infinity}]

Integrate[((u HypergeometricPFQ[{},{(3/2),2},((u^2)/4)])/(-1+(E^u))),{u,0,Infinity}]//Timing

(∑_(n=1)^∞)((ζ(2 n))/((2 n)!))

Numeric

NSum[Zeta[2n]/(2n)!,{n,∞},WorkingPrecision->50]

v5

Sum[1/(2n)!/Gamma[2n]u^(2n-1)/(E^u-1),{n,2,∞}]

FullSimplify[%,u>=0]

Integrate[((-(u^2)+(Sqrt[u]) BesselI[1,2 (Sqrt[u])]-(Sqrt[u]) BesselJ[1,2 (Sqrt[u])])/(2 (-1+(E^u)) u)),{u,0,∞}]

Integrate[(1/(2-2 (E^u)))(u+Hypergeometric0F1Regularized[2,-u]-Hypergeometric0F1Regularized[2,u]),{u,0,∞}]

v6

Sum[Zeta[2n]/(2n)!,{n,∞}]

Integrate[((-(u^2)+(Sqrt[u]) BesselI[1,2 (Sqrt[u])]-(Sqrt[u]) BesselJ[1,2 (Sqrt[u])])/(2 (-1+(E^u)) u)),{u,0,∞}]//Timing

Integrate[(1/(2-2 (E^u)))(u+Hypergeometric0F1Regularized[2,-u]-Hypergeometric0F1Regularized[2,u]),{u,0,∞}]//Timing

Not in OEIS

Connection with Bernoulli Numbers

Table[{n,BernoulliB[n],(-1)^(n+1)n  Zeta[1-n],-n Zeta[1-n]},{n,15}]//ColumnForm

Table[-n Zeta[1-n],{n,10}]

Table[BernoulliB[n],{n,10}]

ζ(2 n)==((((((-1))^(n+1)) n (2 n-1) (2^(2 n-2)) (π^(2 n))) (∫_0^1)(Subscript[E, 2 (n-1)])(x)dx)/(((2^(2 n))-1) (2 n)!))

(((((-1))^(n+1))n(2n-1)(2^(2n-2))(π^(2n)))/(((2^(2n))-1)(2n)!))(∫_0^1)EulerE[2(n-1),x]\[DifferentialD]x

Table[(((((-1))^(1+n)) (2^(-2+2 n)) n (-1+2 n) (π^(2 n)) (-((EulerE[-1+2 n,0])/(-1+2 n))+((EulerE[-1+2 n,1])/(-1+2 n))))/((-1+(2^(2 n))) (2 n)!))-Zeta[2n],{n,10}]

ζ(2 n+1)==((((((-1))^n) (π^(2 n+1))) (∫_0^1)(Subscript[E, 2 n])(x) tan(((π x)/2))dx)/(4 (1-(2^(-2 n-1))) (2 n)!))

J. Crepps

(((((-1))^n) (π^(2 n+1)))/(4(1-(2^(-2n-1))) (2 n)!))(∫_0^1)EulerE[2n,x]Tan[(π/2)x]\[DifferentialD]x

Table[(((((-1))^n) (π^(2 n+1)))/(4(1-(2^(-2n-1))) (2 n)!))(∫_0^1)EulerE[2n,x]Tan[(π/2)x]\[DifferentialD]x-Zeta[2n+1],{n,5}]//Timing

ζ(2 n+1)==((((((-1))^n) (π^(2 n+1))) (∫_0^1)(Subscript[E, 2 n])(x) cot(((π x)/2))dx)/(4 (1-(2^(-2 n-1))) (2 n)!))

J. Crepps

(((((-1))^n) (π^(2 n+1)))/(4(1-(2^(-2n-1))) (2 n)!))(∫_0^1)EulerE[2n,x]Cot[(π/2)x]\[DifferentialD]x

Table[(((((-1))^n) (π^(2 n+1)))/(4(1-(2^(-2n-1))) (2 n)!))(∫_0^1)EulerE[2n,x]Cot[(π/2)x]\[DifferentialD]x-Zeta[2n+1],{n,5}]//Timing

ζ(2 n+1)==((((((-1))^n) (2^(2 n)) (π^(2 n+1))) (∫_0^1)(Subscript[B, 2 n+1])(x) tan(((π x)/2))dx)/((2 n+1) (2 n)!))

J. Crepps

(((((-1))^n)(2^(2n))(π^(2n+1)))/((2n+1)(2n)!))(∫_0^1)BernoulliB[2n+1,x]Tan[(π/2)x]\[DifferentialD]x

Table[(((((-1))^n)(2^(2n))(π^(2n+1)))/((2n+1)(2n)!))(∫_0^1)BernoulliB[2n+1,x]Tan[(π/2)x]\[DifferentialD]x-Zeta[2n+1],{n,5}]//Timing

ζ(2 n+1)==((((((-1))^(n+1)) (2^(2 n)) (π^(2 n+1))) (∫_0^1)(Subscript[B, 2 n+1])(x) cot(((π x)/2))dx)/((2 n+1) (2 n)!))

J. Crepps

(((((-1))^(n+1))(2^(2n))(π^(2n+1)))/((2n+1)(2n)!))(∫_0^1)BernoulliB[2n+1,x]Cot[(π/2)x]\[DifferentialD]x

Table[(((((-1))^(n+1))(2^(2n))(π^(2n+1)))/((2n+1)(2n)!))(∫_0^1)BernoulliB[2n+1,x]Cot[(π/2)x]\[DifferentialD]x-Zeta[2n+1],{n,5}]//Timing

ζ (0)

FullSimplify[Sum[(((-1))^k) Binomial[n,k](((k+1))^(-z))/.z->0,{k,0,n}]]//TraditionalForm

Sum[((KroneckerDelta[-n])/(2^(n+1))),{n,0,∞}]

(1/(1-(2^(1-z))))1/2/.z->0

ζ((1/2))

Zeta[1/2]//FunctionExpand

N[Zeta[1/2],20]

ζ (-1)

FullSimplify[Sum[(((-1))^k) Binomial[n,k](((k+1))^(-z))/.z->-1,{k,0,n}]]//TraditionalForm

Sum[((KroneckerDelta[-n]-n KroneckerDelta[1-n])/(2^(n+1))),{n,0,∞}]

(1/(1-(2^(1-z))))(1/4)/.z->-1

Derivatives

FunctionExpand[Zeta'[-2n],n∈Integers]

Subtract@@@FunctionExpand[Table[{(((2n)!Zeta[2n+1])/((2^(2n+1))(π^(2n))))(((-1))^n),Zeta'[-2n]},{n,10}]]//Union

ζ'(-2n)

Table[Zeta'[-2n],{n,5}]//FunctionExpand

Table[((Zeta[2n+1](2n)!)/((2^(2n+1))(π^(2n))))(((-1))^n),{n,5}]

Through[{Numerator,Denominator}[Table[(((2n)!)/(2^(2n+1)))(((-1))^n),{n,10}]]]

ζ'(-1)

Zeta'[-1]//FunctionExpand

ζ'(s)

-Sum[Log[k]/k^s, {k, Infinity}]

-Sum[Log[k]/k^s,{k,2,Infinity}]

Table[Zeta'[n],{n,-3,3}]

(ζ^((n)))(0)

<<Utilities`Typesetting`

{#,N[#]}&[Zeta'[0]]

D[Zeta[x],x]/.x->0.

FunctionExpand[Zeta''[0]]//FullSimplify//TraditionalForm

FunctionExpand[Zeta'''[0]]//Simplify//TraditionalForm

FunctionExpand[Zeta''''[0]]//FullSimplify//TraditionalForm

(ζ^((n)))((1/2))

n==1

expr1=D[Gamma[s/2]Pi^(-s/2)Zeta[s],s]/.s->1/2

expr2=D[Gamma[(1-s)/2]Pi^(-(1-s)/2)Zeta[1-s],s]/.s->1/2

Simplify[expr1-expr2]

Solve[expr1==expr2,Zeta'[1/2]]//Simplify

Solve[expr1==expr2,Zeta'[1/2]]//FullSimplify

N[(1/4) (2 EulerGamma+π+2 Log[8 π]) Zeta[(1/2)],20]

n==2

expr1=D[Gamma[s/2]Pi^(-s/2)Zeta[s],{s,2}]/.s->1/2

expr2=D[Gamma[(1-s)/2]Pi^(-(1-s)/2)Zeta[1-s],{s,2}]/.s->1/2

Simplify[expr1-expr2]

Solve[expr1==expr2,Zeta''[1/2]]

solve[expr1==expr2/.Zeta''[1/2]->z,z]

ζ'(2)

Solve[Exp[-zeta2]==(((((Glaisher^12)/(2π Exp[EulerGamma]))))^((π^2)/6)),zeta2]//FullSimplify

N[{(1/6) (π^2) (EulerGamma+Log[2]-12 Log[Glaisher]+Log[π]),Zeta'[2]},20]

Zeta'[2]//FunctionExpand

Khinchin's Constant

Zeta'[s_]:=Module[{eps=1. 10^(-10),sn=N[s]},
  (Zeta[sn+eps]-Zeta[sn])/eps
]

Zeta'[s_,a_]:=Module[{eps=1. 10^(-10),an=N[a],sn=N[s]},
  (Zeta[sn+eps,an]-Zeta[sn,an])/eps
]

K = 2.685452001

K1[n_]:=Exp[1/Log[2.] Sum[(-1)^j(2-2^j)Zeta'[j]/j,{j,2,n}]]//N

K1List=N[Table[K1[n],{n,10,40,5}],10]

K2[n_]:=Exp[1/Log[2.] Sum[-Log[k+1](Log[k+3]-2Log[k+2]+Log[k+1])-
  (-1)^k(2-2^(k+2))(Log[k+1]/(k+1)^(k+2)-Zeta'[k+2,k+2])/(k+2)+
  Log[k+1] Sum[(-1)^s(2-2^s)/((k+1)^s s),{s,1,k+2}],{k,0,n}]//N
]

K2List=N[Table[K2[n],{n,1,10}],10]

N[K2[20],10]

PrimeZeta[n_]:=NSum[MoebiusMu[k]/k Log[Zeta[k n]],{k,Infinity},Method->Fit]

Table[PrimeZeta[i],{i,2,5}]

Plouffe's Sum Identities

S[m_]:=Module[{p=Abs[m],s=Sign[m]},
		Sum[1/(k^p(Exp[2Pi k]+s)),{k,Infinity}]
]

zeta[k_]:=2^(k-1)Pi^k/(k+1)! Sum[(-1)^(n-1)Binomial[k+1,2n]BernoulliB[k+1-2n]BernoulliB[2n],{n,0,(k+1)/2}]-2 S[k]/; Mod[k,4]==3

zeta[k_]:=(2Pi)^k/(k+1)!/(k-1) Sum[(-1)^n(k+1-4n)Binomial[k+1,2n]BernoulliB[k+1-2n]BernoulliB[2n],{n,0,(k-1)/4}]-2 Sum[(Exp[2Pi n](1+4 Pi n/(k-1))-1)/n^k/(Exp[2Pi n]-1)^2,{n,∞}]/;Mod[k,4]==1 && k≥5

FullSimplify[Table[(zeta[n]+2S[n])/Pi^n,{n,3,21,4}]]

Table[(zeta[n]+2S[n])/Pi^n,{n,3,21,4}]//InputForm

Sum[1/k^n/(Exp[2Pi k]-1),{k,∞}]

Simplify[Table[zeta[n],{n,5,11,4}]]

Simplify[Apart[(((E^(2 π n)) (1+((4 π n)/(5-1)))-1)/((n^5) ((((E^(2 π n))-1))^2)))]]

N[%,20]

N[Table[Zeta[n],{n,5,21,4}],20]

zeta[3]:=7Pi^3/180-2S[-3]
zeta[5]:=Pi^5/294- 2/35S[5]-72/35S[-5]
zeta[7]:=19Pi^7/56700-2S[-7]
zeta[9]:=125Pi^9/3704778-992S[-9]/495-2S[9]/495
zeta[11]:=1453Pi^11/425675250-2S[-11]
zeta[13]:=89/257432175Pi^13-16512/8255 S[-13]-2/8255S[13]
zeta[15]:=13687Pi^15/390769879500-2S[-15]
zeta[17]:=397549Pi^17/112024529867250-261632S[-17]/130815-2/130815S[17]
zeta[19]:=7708537Pi^19/21438612514068750-2S[-19]
zeta[21]:=((68529640373 (π^21))/1881063815762259253125)-((4196352 S[-21])/2098175)-((2 S[21])/2098175)

ExpandAll[(89Pi^13-514926720s[-13]-62370s[13])/257432175]

ExpandAll[%198[[1,1,2]]]

Table[{i,N[zeta[i]-Zeta[i]]},{i,3,21,2}]//TableForm

Riemann Zeta Inverse (Power Free)

<<Utilities`Typesetting`

Plot[1/Zeta[n],{n,1,10},
AxesLabel->TraditionalForm/@{x,InlineFraction[1,Zeta[x]]},
Epilog->
	{Table[{Hue[(i-2)/9.],
Line[{{i,0},{i,1/Zeta[i]},{0,1/Zeta[i]}}]
},{i,2,10}],
Table[{Hue[(i-2)/9.],
PointSize[.03],Point[{i,1/Zeta[i]}]
},{i,2,10}]
}
]

TableForm[Flatten/@Transpose[{
Table[{n,1./Zeta[n]},{n,2,6}],
Table[Count[Range[10^k],_?(PowerfreeQ[n,#]&)],{n,2,6},{k,1,6}]
}],TableAlignments->Right]