Porter's Constant

Author

Eric W. Weisstein
November 21, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Numerical

Numerical

N[p1,110]

N[p2,110]

Closed Forms

((6 log(2) (48 log(A)-2-log(2)-4 log(π)))/(π^2))-(1/2)

p1=((6Log[2](48Log[Glaisher]-Log[2]-4Log[π]-2))/(π^2))-(1/2);

((6Log[2](48Log[Glaisher]-Log[2]-4Log[π]-2))/(π^2))-(1/2)//N

RealDigits[p1,10,110][[1]]

((6 log(2) ((π^2) (4 ℽ-2+log(8))-24 ζ'(2)))/(π^4))+(1/2)

p2=((6 Log[2] ((π^2) (-2+4 EulerGamma+Log[8])-24 Zeta'[2]))/(π^4))-(1/2);

FullSimplify[p1-p2]

FullSimplify/@{p1,p2}

Porter

B

NSum[(1/i)(Sum[(1/j),{j,i+1,2i-1}]-Log[2]),{i,∞}]

Sum[(1/j),{j,i+1,2i-1}]-Log[2]//FullSimplify

$Version

Sum[(1/i)(EulerGamma-HarmonicNumber[i]-Log[2]+PolyGamma[0,2 i]),{i,∞}]//Head

bug(342905): Porter constant-related Sum[(EulerGamma - HarmonicNumber[i] - Log[2] + PolyGamma[0, 2 i])/i, {i, Infinity}] incorrectly reported to diverge

Plot[((EulerGamma-HarmonicNumber[i]-Log[2]+PolyGamma[0,2 i])/i),{i,1,10}]

NSum[(1/i)(EulerGamma-HarmonicNumber[i]-Log[2]+PolyGamma[0,2 i]),{i,∞}]

NSum[(1/k)(HarmonicNumber[2k-1]-HarmonicNumber[k]-Log[2]),{k,∞}]

Sum[(1/k)(HarmonicNumber[2k-1]-HarmonicNumber[k]-Log[2]),{k,∞}]//Head

((Log[2])^2)-((π^2)/6)//N

bug(342904): Porter constant-related sums Sum[(EulerGamma - HarmonicNumber[i] - Log[2] + PolyGamma[0, 2 i])/i, {i, Infinity}] and Sum[(HarmonicNumber[2 k - 1] - HarmonicNumber[k] - Log[2])/k, {k, Infinity}] could return Log[2]^2 - Pi^2/6

I

ii=NIntegrate[(((t^2)Log[t])/(Sqrt[(t^2)+1])),{t,0,(1/(2(Sqrt[2])))}]

(1/96)(2(π^2)+30((Log[2])^2)-39Log[2]-9)//N

Integrate[(((t^2)Log[t])/(Sqrt[(t^2)+1])),{t,0,(1/(2(Sqrt[2])))}]//Timing

bug(342908): Porter constant-related integral Integrate[(t^2 Log[t])/Sqrt[t^2 + 1], {t, 0, 1/(2 Sqrt[2])}] could (quickly) return simpler answer (2 Pi^2 + 30 Log[2]^2 - 39 Log[2] - 9)/96

FullSimplify[FunctionExpand@Last[%81]]

N[%]

Moebius sum

sum2=Sum[((MoebiusMu[l]Log[l])/(l^2)),{l,∞}]

Sum[((MoebiusMu[l]Log[l])/(l^2)),{l,∞}]

FullSimplify[%]

FullSimplify[(6/(π^2))Zeta'[2]]

FunctionExpand[Glaisher]

Total

(6/(π^2))(b+(3/16)+(EulerGamma+(5/16))Log[2]-(7/8)((Log[2])^2)+2ii)-Log[2]sum2

From Knuth:

(p=-(3/4)+(6/(π^2))((3/4)((Log[2])^2)+(EulerGamma-(1/2))Log[2]-(6/(π^2))Zeta'[2]Log[2]))//N

FullSimplify[p]

Knuth

N[p=-(3/4)+(6/(π^2))((3/4)((Log[2])^2)+(EulerGamma-(1/2))Log[2]-(6/(π^2))Zeta'[2]Log[2]),12]

N[c=((6 Log[2] ((π^2) (-2+4 EulerGamma+Log[8])-24 Zeta'[2]))/(π^4))-(1/2),12]

FullSimplify[c-p]

p//FullSimplify//TraditionalForm

c//FullSimplify//TraditionalForm