Murata's Constant

Author

Eric W. Weisstein
December 9, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Value

Times@@(1`20+(1/(((#-1))^2))&[Prime[Range[(10^6)]]])//Timing

Times@@(1`20+(1/(((#-1))^2))&[Prime[Range[(10^7)]]])//Timing

Product[1+(1/(((Prime[n]-1))^2)),{n,∞}]

NProduct[1+(1/(((Prime[n]-1))^2)),{n,∞}]

Times Another way

prod=N[1,25];
(Monitor[Do[prod*=With[{p=Prime[k]},1+(1/(((p-1))^2))],{k,10^6}],{k,prod}];prod)//Timing

Series

Series[Log[1+(1/(((p-1))^2))],{p,∞,10}]

SeriesCoefficient[Log[1+(1/(((p-1))^2))],{p,∞,n}]//FullSimplify

Series[x/(1-2*x+2*x^2),{x,0,10}]

SeriesCoefficient[x/(1-2*x+2*x^2),{x,0,n}]

ComplexExpand[Im[(1+I)^n]]

Table[(2^(n/2)) Sin[((n π)/4)],{n,3,22}]

Table[(((1-I))^n)+(((1+I))^n),{n,20}]

Table[((-2+(((1-I))^n)+(((1+I))^n))/n),{n,10}]

FullSimplify[(2^(n/2)) Sin[((n π)/4)]/.n->n+2]

Table[-((-2+((2^(1+(n/2))) Cos[((n π)/4)]))/n),{n,10}]

FullSimplify[-((-2+((2^(1+(n/2))) Cos[((n π)/4)]))/n)]

Exp[Sum[-((-2+(((1-I))^n)+(((1+I))^n))/n)PrimeZetaP[n],{n,2,50}]]//N

Exp[Sum[((2-((2^(1+(n/2))) Cos[((n π)/4)]))/n)PrimeZetaP[n],{n,2,50}]]//N

Exp[Sum[(2/n)PrimeZetaP[n],{n,2,50}]-Sum[((((2^(1+(n/2))) Cos[((n π)/4)]))/n)PrimeZetaP[n],{n,2,50}]]//N

Sum[(2/n)PrimeZetaP[n],{n,2,∞}]

Sum[-((-2+(((1-I))^n)+(((1+I))^n))/n)PrimeZetaP[n],{n,∞}]

Block[{$MaxExtraPrecision=1000},Accumulate[Table[-((-2+(((1-I))^n)+(((1+I))^n))/n)PrimeZetaP[n]//N[#,50]&,{n,2,20}]]]//Chop

Exp[Last[%]]

PrimeZetaP from Series

ser=Series[Log[1+(1/(((p-1))^2))],{p,∞,2010}][[3]];

Block[{$MaxExtraPrecision=1000},
sums=Exp[Accumulate[N[Table[ser[[k]] PrimeZetaP[k+1],{k,2000}],400]]]
];//Timing

ListPlot[Log10[Abs[Last[sums]-sums]]]

sums[[-1]]

Abs[sums[[-1]]-sums[[-2]]]

N[Last[sums],-Log10[Abs[sums[[-1]]-sums[[-2]]]]-2]

-Log10[Abs[sums[[-1]]-sums[[-2]]]]-2