A255703 - OEIS

%I #27 Jun 05 2023 02:01:59

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

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

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

%N Decimal expansion of the Plouffe sum S(5,4) = Sum_{n >= 1} 1/(n^5*(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=5, m=4 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(4*Pi*k)). - _Amiram Eldar_, Jun 05 2023

%e 0.000003487354897856938267595604056105856171160604720764172...

%t digits = 102; S[5, 4] = NSum[1/(n^5*(Exp[4*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[5, 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)), A255700 (S(3,4)), A255701 (S(5,1)), A255702 (S(5,2)).

%Y Cf. A001160 (sigma_5), A013663 (zeta(5)).

%K nonn,cons,easy

%O -5,1

%A _Jean-François Alcover_, Mar 02 2015