Lévy Constant

Author

Eric W. Weisstein
July 10, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Initialization

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

KhinchinLevy[x_,n_:200]:=(Denominator[FromContinuedFraction[Rest[ContinuedFraction[x,n+1]]]])^(1/n)

Value

(KL=Exp[Pi^2/(12Log[2])])//N

Pi

KhinchinLevy[Pi,10000]//N//Timing

γ

KhinchinLevy[EulerGamma,10000]//N//Timing

Integral

(i=Integrate[((Log[(x^(-1))])/((x+1)Log[2])),{x,0,1}])//Timing

<<Utilities`Simplify`

i//LogExpand

Plot

CopelandErdosConstant=FromDigits[{Flatten[IntegerDigits/@Table[Prime[n],{n,1000}]],0}];

ListPlot[
KhinchinLevyList[#,500]&/@{Pi,Sin[1],EulerGamma,CopelandErdosConstant},
Joined->False,
PlotStyle->{Red,Green,Blue,Yellow},PlotLegends->{Pi,Sin[1],EulerGamma,C},
Epilog->{Line[{{-10,KL},{1000,KL}}]},
AxesLabel->{n,Row[{Subsuperscript[q,n,Row[{1,"/",n}]]}]},LabelStyle->Directive[Medium,Black,FontFamily->"Times"]]