Hafner-Sarnak-McCurley Constant

Author

Eric W. Weisstein
April 16, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Hafner-Sarnak-McCurleyConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Hafner-Sarnak-McCurleyConstant.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Definition

d[n_,nmax_]:=Product[1-(1-Product[1-Prime[k]^(-j),{j,n}])^2,{k,nmax}]
d[n_]:=d[n,Infinity]

Values

D(1)

(1-Product[1-Prime[k]^(-j),{j,1}])^2

Product[1-(1/((Prime[k])^2)),{k,∞}]

d[1]

Product[1-(1-Product[1-Prime[k]^(-j),{j,1}])^2,{k,∞}]

D(2)

d[2]

N[d[2],100]//FullForm

D(∞)

σ=SetPrecision[0.3532363719,10]

N[d[100,100],20]//Timing

N[d[100,200],20]//Timing

Table

Table[N[d[n,500]],{n,10}]//Timing

(ds=Table[N[d[n,1000]],{n,10}])//Timing

(ds=Table[N[d[n,2000]],{n,10}])//Timing

(ds=Table[N[d[n,3000]],{n,10}])//Timing

Plot

MakeBoxes[ordinate,TraditionalForm]:=RowBox[{"D","(","n",")"}]

ListPlot[ds,
AxesLabel->TraditionalForm/@{n,ordinate},
Joined->True,PlotStyle->Red,
Epilog->{Red,PointSize[.02],MapIndexed[Point[{#2[[1]],#1}]&,ds]},
AxesOrigin->{1,.3},
PlotRange->{{.9,10.1},{.3,.65}}]