A258945 Decimal expansion of Dickman's constant C_4.

0, 6, 7, 6, 4, 5, 2, 0, 2, 1, 0, 6, 9, 4, 6, 1, 3, 6, 9, 6, 9, 7, 5, 0, 2, 3, 1, 0, 3, 3, 8, 2, 2, 9, 9, 3, 9, 2, 3, 4, 2, 1, 9, 3, 4, 4, 9, 4, 9, 2, 0, 4, 3, 1, 7, 3, 0, 1, 8, 6, 0, 1, 3, 4, 6, 5, 2, 5, 7, 5, 3, 8, 5, 1, 1, 6, 8, 5, 5, 5, 2, 9, 0, 9, 8, 0, 6, 8, 5, 2, 2, 4, 6, 3, 5, 6, 2, 4, 4, 8, 2, 7, 0, 6

Offset: 0

Author

Jean-François Alcover, Jun 15 2015

Keywords

• nonn
• cons

Examples

			0.067645202106946136969750231033822993923421934494920431730186...
		

Links

• David Broadhurst, Dickman polylogarithms and their constants arXiv:1004.0519 [math-ph], 2010.

Crossrefs

Cf. A175475, A245238.

Programs

Mathematica

digits = 103; C4 = NIntegrate[(Log[x/(2*x+1)]*PolyLog[2, x] + (1/2)*Log[x]^2* PolyLog[1, -2*x])/(x*(x+1)), {x, 0, 1/2}, WorkingPrecision -> digits+5] + 3*PolyLog[4, 1/2] - 3/8 *PolyLog[4, 1/4] - 3/4* Log[2]*PolyLog[3, 1/4] + (Pi^2 - 9*Log[2]^2)/12*PolyLog[2, 1/4] + 21*Log[2]*Zeta[3]/8 + Pi^2*(Log[2]^2/24) - Pi^2*Log[2]*(Log[3]/6) + Log[2]^3*Log[3]/2 - 5*Log[2]^4/8; Join[{0}, RealDigits[C4, 10, digits] // First]

Python

from mpmath import mp, log, polylog, zeta, pi, quad
mp.dps=104
f=lambda x: (log(x/(2*x+1))*polylog(2, x) + (1/2)*log(x)**2*polylog(1, -2*x))/(x*(x+1))
I=quad(f, [0, 1/2]) + 3*polylog(4, 1/2) - 3/8*polylog(4, 1/4) - 3/4*log(2) * polylog(3, 1/4) +(pi**2 - 9*log(2)**2)/12*polylog(2, 1/4) + 21*log(2)*zeta(3)/8 + pi**2*(log(2)**2/24) - pi**2*log(2)*(log(3)/6) + log(2)**3*log(3)/2 - 5*log(2)**4/8
print([int(z) for z in list(str(I)[2:-1])]) # Indranil Ghosh, Jul 03 2017

Formula

C_1 = 0, C_2 = -Pi^2/12, C_3 = -zeta(3)/3.

C_4 = Integral_{0..1/2} (log(x/(2*x+1))*polylog(2, x) + (1/2)*log(x)^2*polylog(1, -2*x))/(x*(x+1)) dx + 3*polylog(4, 1/2) - 3/8*polylog(4, 1/4) - 3/4*log(2) * polylog(3, 1/4) +(Pi^2 - 9*log(2)^2)/12*polylog(2, 1/4) + 21*log(2)*zeta(3)/8 + Pi^2*(log(2)^2/24) - Pi^2*log(2)*(log(3)/6) + log(2)^3*log(3)/2 - 5*log(2)^4/8.

Also (conjecturally) equals Pi^4/1440.