Apéry's Constant Approximations

Author

Eric W. Weisstein
April 15, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/AperysConstantApproximations.nb.

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

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

Exact

N[Zeta[3],10]

Galliani

Paolo Galliani, 19 Apr 2002

(g=FullSimplify[EulerGamma^(-1/3)+1/(Pi^4((1+2EulerGamma-2/(130+Pi^2))^3))])//TraditionalForm

CorrectDigits[{g,Zeta[3]}]

Pegg

Mon Nov  8

v5.0

RealDigits[Subtract@@{10+Zeta[16]-Sqrt[96],Zeta[3]},10,100]

v5.1

CorrectDigits[{10+Zeta[16]-Sqrt[96],Zeta[3]}]

Hudson

Mon, 08 Nov 2004

CorrectDigits[8 - Log[896],Zeta[3]]

CorrectDigits[(Pi+Pi^2)^(69/962) ,Zeta[3]]

CorrectDigits[(((71/47 + EulerGamma))^(1/4)) ,Zeta[3]]

CorrectDigits[GoldenRatio - 1307/3142 ,Zeta[3]]

CorrectDigits[(19963/4265 + GoldenRatio)^(1/10) ,Zeta[3]]

CorrectDigits[525587^(5123^(-1/2)) ,Zeta[3]]

Lima

CorrectDigits[-(5/197)+(11/394)(π^2)Log[2]-(283/394)π ((Log[2])^2)+(236/197)((Log[2])^3)+(209/394)((Log[1+(Sqrt[2])])^3)+(93/197)π Catalan,Zeta[3]]

v={1,(π^2)Log[2],((Log[2])^2),((Log[2])^3),((Log[1+(Sqrt[2])])^3),π Catalan,Zeta[3]};

Using M--

10

r=FindIntegerNullVector[N[v,10]]

Expand[Solve[(v.r/.Zeta[3]->𝒜)==0,𝒜]][[1,1]]//TraditionalForm

N[v.r,20]//N

20

r=FindIntegerNullVector[N[v,20]]

Expand[Solve[(v.r/.Zeta[3]->𝒜)==0,𝒜]][[1,1]]//TraditionalForm

N[v.r,50]//N

30

r=FindIntegerNullVector[N[v,30]]

Expand[Solve[(v.r/.Zeta[3]->𝒜)==0,𝒜]][[1,1]]//TraditionalForm

N[v.r,50]//N