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%I #19 Oct 10 2020 05:00:11
%S 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,
%T 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,
%U 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
%N Decimal expansion of Dickman's constant C_4.
%H David Broadhurst, <a href="http://arxiv.org/abs/1004.0519">Dickman polylogarithms and their constants</a> arXiv:1004.0519 [math-ph], 2010.
%F C_1 = 0, C_2 = -Pi^2/12, C_3 = -zeta(3)/3.
%F 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.
%F Also (conjecturally) equals Pi^4/1440.
%e 0.067645202106946136969750231033822993923421934494920431730186...
%t 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]
%o (Python)
%o from mpmath import mp, log, polylog, zeta, pi, quad
%o mp.dps=104
%o f=lambda x: (log(x/(2*x+1))*polylog(2, x) + (1/2)*log(x)**2*polylog(1, -2*x))/(x*(x+1))
%o 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
%o print([int(z) for z in list(str(I)[2:-1])]) # _Indranil Ghosh_, Jul 03 2017
%Y Cf. A175475, A245238.
%K nonn,cons
%O 0,2
%A _Jean-François Alcover_, Jun 15 2015
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