%I #24 Jun 05 2023 02:21:03

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

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

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

%N Decimal expansion of the Plouffe sum S(3,4) = Sum_{n >= 1} 1/(n^3*(exp(4*Pi*n)-1)).

%H Steven R. Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/inspired2.pdf">Identities inspired by Ramanujan Notebooks (part 2)</a>, April 2006.

%H Linas Vepštas, <a href="https://doi.org/10.1007/s11139-011-9335-9">On Plouffe's Ramanujan identities</a>, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; <a href="https://cyberleninka.org/article/n/534457.pdf">alternative link</a>; <a href="https://arxiv.org/abs/math/0609775">arXiv preprint</a>, arXiv:math/0609775 [math.NT], 2006-2010.

%F This is the case k=3, m=4 of S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).

%F Pi^3 = 720*S(3,1) - 900*S(3,2) + 180*S(3,4).

%F zeta(3) = 28*S(3,1) - 37*S(3,2) + 7*S(3,4).

%F Equals Sum_{k>=1} sigma_3(k)/(k^3*exp(4*Pi*k)). - _Amiram Eldar_, Jun 05 2023

%e 0.000003487356038004276054514730322548976264651146827033884525679...

%t digits = 104; S[3, 4] = NSum[1/(n^3*(Exp[4*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[3, 4], 10, digits] // First

%Y Cf. A255695 (S(1,1)), A084254 (S(1,2)), A255697 (S(1,4)), A255698 (S(3,1)), A255699 (S(3,2)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).

%Y Cf. A001158 (sigma_3), A002117 (zeta(3)).

%K nonn,cons,easy

%O -5,1

%A _Jean-François Alcover_, Mar 02 2015