OFFSET
0,1
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 80.
LINKS
Eric Weisstein's MathWorld, Lattice Sum
FORMULA
gamma_3 = (1/8)*(delta_3 + 3*(- Pi/6 + log((sqrt(3) + 1)/(sqrt(3) - 1))) - 12*gamma_2 - 6*EulerGamma).
EXAMPLE
0.58174804565972267655489926584685317714602246563144492431364...
MATHEMATICA
digits = 100; k0 = 10; dk = 10; Clear[s]; s[k_] := s[k] = 7*(Pi/6) - 19/2*Log[2] + 4*Sum[(3 + 3*(-1)^m + (-1)^(m + n))*Csch[Pi*Sqrt[m^2 + n^2]]/Sqrt[m^2 + n^2], {m, 1, k}, {n, 1, k}] // N[#, digits + 10] &; s[k0]; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits + 5][[1]] != RealDigits[s[k - dk], 10, digits + 5][[1]], k = k + dk]; Pi0 = s[k]; delta2 = 2*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]); delta3 = Pi0 + Pi/6; gamma2 = (1/4)*(delta2 + 2*Log[(Sqrt[2] + 1)/(Sqrt[2] - 1)] - 4*EulerGamma); gamma3 = (1/8)*(delta3 + 3*(- Pi/6 + Log[(Sqrt[3] + 1)/(Sqrt[3] - 1)]) - 12*gamma2 - 6*EulerGamma); RealDigits[gamma3, 10, 102] // First
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Sep 11 2014
STATUS
approved