Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #21 Jun 05 2023 02:01:54
%S 1,8,7,1,0,4,5,6,0,5,3,0,1,2,5,9,5,1,4,8,7,3,9,5,1,4,7,5,8,1,0,5,6,3,
%T 4,3,0,3,1,8,9,6,2,8,2,3,0,8,7,5,8,2,8,6,5,6,0,4,2,4,5,2,7,9,8,5,5,2,
%U 5,8,3,5,4,0,9,5,1,0,4,2,3,0,2,7,7,5,4,9,7,6,1,1,3,0,7,4,8,9,6,9,7,9,3,6
%N Decimal expansion of the Plouffe sum S(5,2) = Sum_{n >= 1} 1/(n^5*(exp(2*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=5, m=2 of S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).
%F zeta(5) = 24*S(5,1) - (259/10)*S(5,2) - (1/10)*S(5,4).
%F Equals Sum_{k>=1} sigma_5(k)/(k^5*exp(2*Pi*k)). - _Amiram Eldar_, Jun 05 2023
%e 0.0018710456053012595148739514758105634303189628230875828656...
%t digits = 104; S[5, 2] = NSum[1/(n^5*(Exp[2*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[5, 2], 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)), A255700 (S(3,4)), A255701 (S(5,1)), A255703 (S(5,4)).
%Y Cf. A001160 (sigma_5), A013663 (zeta(5)).
%K nonn,cons,easy
%O -2,2
%A _Jean-François Alcover_, Mar 02 2015