Glaisher-Kinkelin Constant

Author

Eric W. Weisstein
January 6, 2019

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Glaisher-KinkelinConstant.nb.

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

©2019 Wolfram Research, Inc. except for portions noted otherwise

Notation

Glaisher//TraditionalForm

Limits

(lim_(n->∞)) ((H(n))/((n^(((n^2)/2)+(n/2)+(1/12))) (e^(-((n^2)/4)))))==A

V6

Product[k^k,{k,n}]

Limit[((Product[k^k,{k,n}])/((n^(n^2/2+n/2+1/12))(E^(-n^2/4)))),n->∞]

N[((Product[k^k,{k,n}])/((n^(n^2/2+n/2+1/12))(E^(-n^2/4))))/.n->10^3,20]//Timing

N[Glaisher]

V7

Limit[((Hyperfactorial[n])/((n^(n^2/2+n/2+1/12))(E^(-n^2/4)))),n->∞]

(lim_(n->∞)) ((G(n+1))/((n^((n^2)/2-1/12)) (((2 π))^(n/2)) exp(-((3 (n^2))/4))))==((e^(1/12))/A)

Limit[((BarnesG[n+1])/((n^((n^2)/2-1/12))(((2π))^(n/2))Exp[-3(n^2)/4])),n->∞]

Continued fraction

ContinuedFraction[Glaisher,20]

Incrementally Largest Terms

HighWaterMarks[ContinuedFraction[Glaisher,10^3]]//Timing

HighWaterMarks[ContinuedFraction[Glaisher,2 10^3]]//Timing

HighWaterMarks[ContinuedFraction[Glaisher,5 10^3]]//Timing

HighWaterMarks[ContinuedFraction[Glaisher,7 10^3]]//Timing

HighWaterMarks[ContinuedFraction[Glaisher,10^5]]//Timing

t={};Do[Print[{i,AppendTo[t,First[Timing[ContinuedFraction[Glaisher,i]]]][[-1]]}],{i,100,7000,100}]

Analytic forms

A==(e^((1/12)-ζ'(-1)))

Exp[1/12-Zeta'[-1]]

(((((A^12)/(2 π (e^ℽ)))))^((π^2)/6))==(e^(-ζ'(2)))

FullSimplify[Subtract@@{Exp[-Zeta'[2]],(((((Glaisher^12)/(2π Exp[EulerGamma]))))^((π^2)/6))}]

A==((((e^(((ℽ (π^2))/6)-ζ'(2)))))^(1/(2 (π^2)))) ((2 π)^(1/12))

FullSimplify[((((E^(((EulerGamma (π^2))/6)-Zeta'[2]))))^(1/(2 (π^2)))) (((2 π))^(1/12))]

Integrals

<<Utilities`Simplify`

(∫_0^(1/2))log(Γ(x+1))dx==-(1/24) 7 log(2)+((3 log(A))/2)+((log(π))/4)-(1/2)

Integrate[Log[Gamma[x+1]],{x,0,1/2}]//LogExpand//Expand

(∫_0^(1/2))logΓ(x+1)dx==-(1/24) 7 log(2)+((3 log(A))/2)+((log(π))/4)-(1/2)

Integrate[LogGamma[x+1],{x,0,1/2}]//LogExpand//Expand

Integrate[Log[Gamma[x+1]],{x,0,1/2}]//LogExpand//Expand

(∫_0^∞)((x log(x))/((e^(2 π x))-1))dx==(1/24)-((log(A))/2)

Integrate[((x Log[x])/(Exp[2π x]-1)),{x,0,∞}]

(∫_0^(1/2))logΓ(x)dx==((5 log(2))/24)+((3 log(A))/2)+((log(π))/4)

Integrate[Log[Gamma[x]],{x,0,1/2}]//LogExpand//Expand

Integrate[LogGamma[x],{x,0,1/2}]//LogExpand//Expand

Products

A==(2^(1/36)) (π^(1/6)) (e^((1/3) (s-ℽ/4)))

V6

Sum[(1/(((2k-1))^r)),{k,∞}]

Sum[(((((-(1/2)))^r) (-1+(2^r)) Zeta[r])/(1+r)),{r,2,∞}]//Expand

Solve[s==%/.Glaisher->A,A]

V7

Sum[(1/(((2k-1))^r)),{k,∞}]

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

(∏_(k=1)^∞)(k^(1/(k^2)))==(e^(-(1/6) (ℽ (π^2)))) (A^(2 (π^2))) (((2 π))^(-(π^2)/6))

V6

Product[k^(1/k^2),{k,∞}]

V7

Product[k^(1/k^2),{k,∞}]

FullSimplify[%]

(∏_(k=1, 3,…)^∞)(k^(1/(k^2)))==exp(-(1/24) (π^2) (-36 ln(A)+ln(16)+3 ln(π)+3 ℽ))

Numerical

NProduct[(2k+1)^(1/(2k+1)^2),{k,0,∞},WorkingPrecision->30]

NProduct[k^(1/k^2),{k,1,∞,2},WorkingPrecision->30]

N[(Glaisher^12/(2^(4/3)π E^EulerGamma))^(Pi^2/8),20]

v6

Product[k^(1/k^2),{k,1,∞,2}]

V7

Product[k^(1/k^2),{k,1,∞,2}]

(∏_(k=0)^∞)(((2 k+1))^(1/(((2 k+1))^2)))==(e^(-(1/24) (π^2) (log(16)-36 log(A)+3 log(π)+3 ℽ)))

V6

Product[(2k+1)^(1/(2k+1)^2),{k,0,∞}]

V7

Product[(2k+1)^(1/(2k+1)^2),{k,0,∞}]

(∏_(k=1)^∞)((((4 k+1))^(1/(((4 k+1))^3)))/(((4 k-1))^(1/(((4 k-1))^3))))

Exact

<<Utilities`Simplify`

p=Sum[((Zeta[k])/(k(k+1)(k+2)))(1/(4^k)),{k,3,∞,2}]

p//LogExpand//Expand

N[p]

N[((((Glaisher/((2^(5/32))(π^(1/32))))Exp[-3/32-EulerGamma/48+p/4]))^(π^3)),10]

{
(E^((1/64) (3 π Zeta[3]+(π^3) (3-2 EulerGamma-128 Derivative[1,0][Zeta][-2,(1/4)])))) (((2 π))^(-((π^3)/32))),
 (E^(-Derivative[1][DirichletBeta][3] )),
(2^((π^3)/16)) (E^( (-Derivative[1,0][Zeta][3,(1/4)]+Derivative[1,0][Zeta][3,(3/4)])/64))
}//N

Solve[(E^((1/64) (3 π Zeta[3]+(π^3) (3-2 EulerGamma-128z)))) (((2 π))^(-((π^3)/32)))==Exp[-d],z]

FullSimplify[(1/(48 (π^2)))(27 (π^2)-2 EulerGamma (π^2)+12 (π^2) Log[4]-192 (π^2) Log[Glaisher]+9 Zeta[3]-384 (π^2) z)/.%[[1]],d>0]//LogExpand//Expand

N[(3/8)+(EulerGamma/12)-((4 Derivative[1][DirichletBeta][3])/(π^3))+((5 Log[2])/8)-4 Log[Glaisher]+((Log[π])/8)]

<<Utilities`Simplify`

FullSimplify[p]//LogExpand

FullSimplify[((((Glaisher/((2^(5/32))(π^(1/32))))Exp[-3/32-EulerGamma/48+p/4]))^(π^3))]

FullSimplify[Derivative[1][DirichletBeta][3]]

{
(E^((1/64) (3 π Zeta[3]+(π^3) (3-2 EulerGamma-128 Derivative[1,0][Zeta][-2,(1/4)])))) (((2 π))^(-((π^3)/32))),
 (E^(-Derivative[1][DirichletBeta][3] )),
(2^((π^3)/16)) (E^( (-Derivative[1,0][Zeta][3,(1/4)]+Derivative[1,0][Zeta][3,(3/4)])/64))
}//N

zero=(2^((π^3)/16)) (E^((1/64) (-Derivative[1,0][Zeta][3,(1/4)]+Derivative[1,0][Zeta][3,(3/4)])))-(E^((1/64) (3 π Zeta[3]+(π^3) (3-2 EulerGamma-128 Derivative[1,0][Zeta][-2,(1/4)])))) (((2 π))^(-((π^3)/32)));

N[zero,20]

Developer`ZeroQ[zero]

FullSimplify[zero]

Numeric

NProduct[((((4k+1))^(1/(((4k+1))^3)))/(((4k-1))^(1/(((4k-1))^3)))),{k,∞},WorkingPrecision->10]//Timing

Product[((((4k+1))^(1/(((4k+1))^3)))/(((4k-1))^(1/(((4k-1))^3)))),{k,10}]//N

v5.2

Product[((((4k+1))^(1/(((4k+1))^3)))/(((4k-1))^(1/(((4k-1))^3)))),{k,∞}]

v6

Product[((((4k+1))^(1/(((4k+1))^3)))/(((4k-1))^(1/(((4k-1))^3)))),{k,∞}]

N[%]

V7

Product[((((4k+1))^(1/(((4k+1))^3)))/(((4k-1))^(1/(((4k-1))^3)))),{k,∞}]

FullSimplify[%]

Series

(∑_(k=2)^∞)((log(k))/(k^2))==-(1/6) (π^2) log(2)+2 (π^2) log(A)-(1/6) (π^2) log(π)-((ℽ (π^2))/6)==-ζ'(2)

V6

Sum[((Log[k])/(k^2)),{k,2,∞}]//LogExpand//Expand

FunctionExpand[-Zeta'[2]]//Expand

V7

Sum[((Log[k])/(k^2)),{k,2,∞}]

(∑_(k=1)^∞)((log(2 k+1))/(((2 k+1))^2))==-(1/6) (π^2) log(2)+((ℽ (π^2))/8)+(3/2) (π^2) log(A)-(1/8) (π^2) log(π)

Numeric

NSum[((Log[k])/(k^2)),{k,3,∞,2}]

NSum[((Log[2k+1])/(((2k+1))^2)),{k,∞}]

v6

Sum[((Log[k])/(k^2)),{k,3,∞,2}]//Expand

Sum[((Log[2k+1])/(((2k+1))^2)),{k,∞}]//Expand

N[%]

V7

Sum[((Log[k])/(k^2)),{k,3,∞,2}]//FullSimplify

(∑_(n=0)^∞)(1/(2 (n+1)))(∑_(k=0)^n)(((-1))^(k+1)) (({{n}, {k}})) (((k+1))^2) log(k+1)+(1/8)==log(A)

V6

(1/8)+Sum[(1/(2(n+1)))Sum[(((-1))^(k+1))Binomial[n,k](((k+1))^2)Log[k+1],{k,0,n}],{n,0,∞}]

(1/8)+NSum[(1/(2(n+1)))Sum[(((-1))^(k+1))Binomial[n,k](((k+1))^2)Log[k+1],{k,0,n}],{n,0,∞}]

Block[{$MaxExtraPrecision=100},N[(1/8)+Sum[(1/(2(n+1)))Sum[(((-1))^(k+1))Binomial[n,k](((k+1))^2)Log[k+1],{k,0,n}],{n,0,200}],10]]//Timing

Log[Glaisher]//N

V7

(1/8)+Sum[(1/(2(n+1)))Sum[(((-1))^(k+1))Binomial[n,k](((k+1))^2)Log[k+1],{k,0,n}],{n,0,∞}]//InputForm

N[(1/8)+Sum[(1/(2(n+1)))Sum[(((-1))^(k+1))Binomial[n,k](((k+1))^2)Log[k+1],{k,0,n}],{n,0,200}],20]

Log[Glaisher]//N

(∑_(r=2)^∞)(((((-(1/2)))^r) ((2^r)-1) ζ(r))/(r+1))==(1/12) (36 ln(A)-ln(2)-6 ln(π)+3 ℽ)

s=Sum[(((((-(1/2)))^r)((2^r)-1)Zeta[r])/(1+r)),{r,2,∞}]

(2^(1/36))(π^(1/6))Exp[(-(EulerGamma/4)+s)/3]//FullSimplify

BBP-Type Formulas

<<Utilities`Simplify`

Search

Monitor[Do[If[(b=FindBasis[Glaisher,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,1}],{-3^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,2,2}],{-3^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,3,3}],{-3^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,1}],{-2^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,2,2}],{-2^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,3,3}],{-2^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,0,0},{n,64},{p,1}],{-1,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,0,0},{n,64},{p,2,2}],{-1,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,0,0},{n,64},{p,3,3}],{-1,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,2,2}],{2^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,3,3}],{2^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,2,2}],{3^k,n,p}]; Print["DONE."]

Monitor[Do[If[(b=FindBasis[Glaisher,3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,3,3}],{3^k,n,p}]; Print["DONE."]

Khinchin and Khinchin-Lévy constant

KhinchinList[x_,n_:200]:=MapIndexed[#1^(1/#2[[1]])&,Rest@FoldList[Times,1,Rest[ContinuedFraction[Mod[x,1],n+1]]]]
KhinchinLevyList[x_,n_:200]:=MapIndexed[#1^(1/#2[[1]])&,Denominator/@FromContinuedFraction/@Reverse[NestList[Drop[#,-1]&,Rest[ContinuedFraction[Mod[x,1],n+1]],n]]]

<<Graphics`Colors`
<<Utilities`Typesetting`

With[{k=1000,c=Glaisher},
g1=ListPlot[KhinchinList[c,k],PlotJoined->True,PlotStyle->Red,AxesLabel->TraditionalForm/@{n,((Subscript[a, 1])(Subscript[⋯a, n]))^InlineFraction[1,n]},
DisplayFunction->Identity,
PlotLabel->StyleForm["Khinchin's constant",FontFamily->"Times",FontSlant->"Italic"],
Epilog->{Blue,Dashing[{.02}],Line[{{0,#},{k,#}}]&[2.685452001]}];
g2=ListPlot[KhinchinLevyList[c,k],PlotJoined->True,PlotStyle->Red,AxesLabel->TraditionalForm/@{n,(Subscript[q, n])^InlineFraction[1,n]},
DisplayFunction->Identity,
PlotLabel->StyleForm["Khinchin-Lévy constant",FontFamily->"Times",FontSlant->"Italic"],
Epilog->{Blue,Dashing[{.02}],Line[{{0,#},{k,#}}]&[Exp[Pi^2/12/Log[2]]]}
];
]//Timing

Show[GraphicsArray[{g1,g2}]]