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Decimal expansion of gamma_3, a lattice sum constant, analog of Euler's constant for 3-dimensional lattices.
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%I #6 Feb 16 2025 08:33:23

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

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

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

%N Decimal expansion of gamma_3, a lattice sum constant, analog of Euler's constant for 3-dimensional lattices.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 80.

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/LatticeSum.html"> Lattice Sum</a>

%F gamma_3 = (1/8)*(delta_3 + 3*(- Pi/6 + log((sqrt(3) + 1)/(sqrt(3) - 1))) - 12*gamma_2 - 6*EulerGamma).

%e 0.58174804565972267655489926584685317714602246563144492431364...

%t 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

%Y Cf. A247042 (delta_2), A247043 (gamma_2), A247046 (delta_3).

%K nonn,cons,changed

%O 0,1

%A _Jean-François Alcover_, Sep 11 2014