Gram Series

Author

Eric W. Weisstein
September 9, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PrimeNumbers/GramSeries.nb.

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

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

Formula

GramSeries[x_]:=1+Sum[Log[x]^k/k/k!/Zeta[k+1],{k,Infinity}]
MyRiemannR[x_]:=Sum[MoebiusMu[n]/n LogIntegral[x^(1/n)],{n,∞}]
MyRiemannR[x_,m_]:=Sum[MoebiusMu[n]/n LogIntegral[x^(1/n)],{n,m}]

Plots

Gram Series

V5

Plot[GramSeries[x]-PrimePi[x],{x,0,1000},
AxesLabel->TraditionalForm/@{x,G[x]-PrimePi[x]},
PlotStyle->Red
]//Timing

V7

Plot[RiemannR[x]-PrimePi[x],{x,0,1000},AxesLabel->{x,RiemannR[x]-PrimePi[x]},PlotStyle->Red]

Difference Between Numerical Convergence of G(x) and R(x)

Plot[Evaluate[GramSeries[x]-Append[Table[MyRiemannR[x,10^n],{n,5}],MyRiemannR[x]]],{x,0,20},PlotStyle->{Blue,Green,Yellow,Orange,Red,Black},
PlotRange->All,
AxesLabel->TraditionalForm/@{x,G[x]-R[x]}]//Timing

V5

V7

Plot[Evaluate[RiemannR[x]-Append[Table[MyRiemannR[x,10^n],{n,5}],RiemannR[x]]],{x,0,20},PlotStyle->{Blue,Green,Yellow,Orange,Red,Black},
PlotRange->All,AxesLabel->{x,G[x]-RiemannR[x]}]

Sums

RiemannR[10.]

(∑_n^∞)(((MoebiusMu[n]) li((x^(1/n))))/n)==(RiemannR[x])

Sum[MoebiusMu[n]/n LogIntegral[x^(1/n)], {n, ∞}]

Sum[MoebiusMu[n]/n LogIntegral[x^(1/n)], {n, 10^6}]/.x->10.

1+(∑_k^∞)(((ln^k)(x))/(k k! ζ(k+1)))==(RiemannR[x])

1+Sum[Log[x]^k/k/k!/Zeta[k+1],{k,Infinity}]

1+Sum[Log[x]^k/k/k!/Zeta[k+1],{k,100}]/.x->10.

(4/π) (∑_k^∞)(((((-1))^(k-1)) k (((((ln(x))/(2 π))))^(2 k-1)))/((2 k-1) (BernoulliB[2 k])))==2 (∑_k^∞)(((ln^(2 k-1))(x))/((2 k-1) (2 k-1)! ζ(2 k)))==8 (∑_k^∞)(((((-1))^(k-1)) k (ln^(2 k-1))(x))/((2 k-1) (((2 π))^(2 k)) (BernoulliB[2 k])))==(SuperStar[G])(x)

4/Pi Sum[(-1)^(k-1)k/BernoulliB[2k]/(2k-1)(Log[x]/2/Pi)^(2k-1),{k,Infinity}]

4/Pi Sum[(-1)^(k-1)k/BernoulliB[2k]/(2k-1)(Log[x]/2/Pi)^(2k-1),{k,100}]/.x->10.

2Sum[Log[x]^k/k/k!/Zeta[k+1],{k,1,100,2}]/.x->10.

2Sum[(((Log[x])^(2k-1))/((2k-1)!(2k-1)Zeta[2k])),{k,∞}]

2Sum[(((Log[x])^(2k-1))/((2k-1)!(2k-1)Zeta[2k])),{k,100}]/.x->10.

8Sum[(((((-1))^(k-1))k ((Log[x])^(2k-1)))/(BernoulliB[2k](2k-1)(((2π))^(2k)))),{k,∞}]

8Sum[(((((-1))^(k-1))k ((Log[x])^(2k-1)))/(BernoulliB[2k](2k-1)(((2π))^(2k)))),{k,100}]/.x->10.

Integral Analog

Integrate[(((Log[x])^t)/(t Gamma[t+1]Zeta[t+1])),{t,0,∞}]