Sierpinski Constant

Author

Eric W. Weisstein
October 5, 2008

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

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

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

Value

S[x_]:=EulerGamma+Derivative[1][DirichletBeta][x]/DirichletBeta[x]

Numerical

N[Log[4(π^3)((Exp[2EulerGamma])/((Gamma[(1/4)])^4))]π,20]

S[1]

π S[1`50-10^-10]

Exact

Plot[S[x],{x,0,2},PlotStyle->Red]

v5.2

Limit[S[x],x->1]

v6

Limit[S[x],x->1]

V7

expr=EulerGamma + (4^x*((-4^(-x))*Log[4]*(Zeta[x, 1/4] - Zeta[x, 3/4]) + 
       (Derivative[1, 0][Zeta][x, 1/4] - Derivative[1, 0][Zeta][x, 3/4])/4^x))/
     (Zeta[x, 1/4] - Zeta[x, 3/4]);

Limit[expr, x -> 1]

FullSimplify[%]

exprs={Log[4(π^3)((Exp[2EulerGamma])/((Gamma[(1/4)])^4))],((EulerGamma π-π Log[4]-StieltjesGamma[1,(1/4)]+StieltjesGamma[1,(3/4)])/π)};

FullSimplify[exprs]

FullSimplify[Subtract@@exprs]

N[%]

Plot

With[{nmax=1000,s=2.5849817596},
ListPlot[Rest[FoldList[Plus,0,Table[((SquaresR[2,k])/k),{k,nmax}]]]-Pi Log[Range[nmax]],
Joined->True,
PlotStyle->Red,
AxesLabel->TraditionalForm/@{n,-π Log[n]+Sum[r[k]/k,{k,n}]},
Epilog->{Blue,Line[{{0,s},{10^5,s}}]}]
]