Mertens Constant

Author

Eric W. Weisstein
May 26, 2020

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

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

©2020 Wolfram Research, Inc. except for portions noted otherwise

Prime computation

ps=Prime[Range[10^7]];//Timing

(Subscript[B, 1 ])

ListPlot[Log[1-1/#]+1/#&/@Take[ps,10^3],PlotStyle->Red,Joined->True]

(Subscript[B, 1])==(∑_(k=1)^∞)(ln(1-(1/(Subscript[p, k])))+(1/(Subscript[p, k])))+ℽ

V6

EulerGamma+Sum[Log[1-1/Prime[k]]+1/Prime[k],{k,∞}]

V7

EulerGamma+Sum[Log[1-1/Prime[k]]+1/Prime[k],{k,∞}]//Timing

(Subscript[B, 1])==(lim_(x->∞)) ((∑_n^(PrimePi[x]))(1/(Prime[n]))-ln(ln(x)))

Sum[(1/(Prime[n])),{n,PrimePi[x]}]-Log[Log[x]]

Limit[%,x->∞]

AsymptoticSum[(1/(Prime[n])),{n,PrimePi[x]},x->∞]-Log[Log[x]]

DiscreteLimit[Sum[(1/(Prime[n])),{n,PrimePi[x]}]-Log[Log[x]],x->∞]

Asked Devendra 2020-05-26

With[{x=5.*^5},Sum[(1/(Prime[n])),{n,PrimePi[x]}]-Log[Log[x]]]

With[{x=1.*^6},Sum[(1/(Prime[n])),{n,PrimePi[x]}]-Log[Log[x]]]

Total[1./ps]-Log[Log[ps[[-1]]]]//Timing

Total[1.`20/ps]-Log[Log[ps[[-1]]]]//Timing

(Subscript[B, 1])==ℽ+(∑_(n=2)^∞)(((MoebiusMu[n]) ln(ζ(n)))/n)

N[EulerGamma+Sum[MoebiusMu[n]/n Log[Zeta[n]],{n,2,500}],20]

V12.2

EulerGamma+Sum[MoebiusMu[n]/n Log[Zeta[n]],{n,2,∞}]

With[{digits=50},EulerGamma+NSum[((MoebiusMu[n]Log[Zeta[n]])/n),{n,2,Infinity},WorkingPrecision->digits+10,NSumTerms->3digits]]//Timing

(Subscript[B, 1])==ℽ-(∑_(j=2)^∞)(∑_k^∞)(1/(j ((((Prime[k])))^j)))

EulerGamma-Sum[(1/(j ((Prime[k])^j))),{j,2,∞},{k,∞}]

EulerGamma-Sum[(1/(j ((Prime[k])^j))),{j,2,50},{k,50}]//N

(Subscript[B, 1])=ℽ-(∑_(n=2)^∞)((PrimeZetaP[n])/n)

EulerGamma-Sum[PrimeZetaP[n]/n,{n,2,Infinity}]

EulerGamma-NSum[PrimeZetaP[n]/n,{n,2,Infinity}]

(HyperlinkTemplate[A077761, https://oeis.org/A077761None, https://oeis.org/A077761, HyperlinkActionRecycled, HyperlinkActive, , ])

With[{digits=50},EulerGamma-NSum[PrimeZetaP[n]/n,{n,2,Infinity},WorkingPrecision->digits+10,NSumTerms->3digits]]//Timing

With[{digits=105},EulerGamma-NSum[PrimeZetaP[n]/n,{n,2,Infinity},WorkingPrecision->digits+10,NSumTerms->3digits]]//Timing

With[{digits=200},EulerGamma-NSum[PrimeZetaP[n]/n,{n,2,Infinity},WorkingPrecision->digits+10,NSumTerms->3digits]]//Timing

(Subscript[B, 2])

ListPlot[Log[1-1/#]+1/(#-1)&/@Take[ps,10^3],PlotStyle->Red,Joined->True]

EulerGamma+Plus@@(Log[1-1/#]+1/(#-1))&/@(Take[N[ps,20],#]&/@{10^3,10^4,10^5,10^6})

EulerGamma+Plus@@(Log[1-1/#]+1/(#-1))&/@(Take[N[ps,20],#]&/@{10^7})

EulerGamma+Plus@@(Log[1-1/#]+1/(#-1))&/@(Take[N[ps,30],#]&/@{10^3,10^4,10^5,10^6})

EulerGamma+Plus@@(Log[1-1/#]+1/(#-1.))&/@(Take[ps,#]&/@{10^3,10^4,10^5,10^6})//InputForm

(Subscript[B, 2])==(Underoverscript[ℽ+∑, k, ∞])(ln(1-(1/(Prime[k])))+(1/((Prime[k])-1)))

EulerGamma+Sum[Log[1-(1/(Prime[k]))]+(1/(Prime[k]-1)),{k,∞}]

EulerGamma+Sum[Log[1-(1/(Prime[k]))]+(1/(Prime[k]-1)),{k,100}]//N

(Subscript[B, 2])==(Subscript[B, 1])+(∑_k^∞)(1/((Prime[k]) ((Prime[k])-1)))

Mertens[1]+Sum[(1/(Prime[k](Prime[k]-1))),{k,∞}]

.2614972128+Sum[(1/(Prime[k](Prime[k]-1))),{k,100}]

(Subscript[B, 2])==ℽ+(∑_(n=2)^∞)(((EulerPhi[n]) ln(ζ(n)))/n)

EulerGamma+Sum[((EulerPhi[n]Log[Zeta[n]])/n),{n,2,∞}]

EulerGamma+Sum[((EulerPhi[n]Log[Zeta[n]])/n),{n,2,200}]//N

(Subscript[B, 3])

ps=Prime/@Range[10^6];

EulerGamma+Plus@@Table[Plus@@(Log[#]/#^j&/@Take[N[ps,20],#]),{j,2,10}]&/@{10^4,10^5,10^6}

(Subscript[B, 3])==ℽ+(∑_(j=2)^∞)(∑_(k=1)^∞)((ln((Prime[k])))/((((Prime[k])))^j))

EulerGamma+Sum[((Log[Prime[k]])/((Prime[k])^j)),{j,2,∞},{k,1,∞}]

EulerGamma+Sum[((Log[Prime[k]])/((Prime[k])^j)),{j,2,50},{k,1,50}]//N

Products

(∏_n^(PrimePi[x]))(1-(1/(Prime[n])))~((exp(-ℽ))/(ln(x)))

Product[1-(1/(Prime[n])),{n,PrimePi[x]}]

{Product[1-(1/(Prime[n])),{n,PrimePi[x]}],((Exp[-EulerGamma])/(Log[x]))}/.x->1000.//N