OFFSET
-5,1
LINKS
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.
Simon Plouffe, Identities inspired by Ramanujan Notebooks (part 2), April 2006.
Linas Vepštas, On Plouffe's Ramanujan identities, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; alternative link; arXiv preprint, arXiv:math/0609775 [math.NT], 2006-2010.
FORMULA
This is the case k=3, m=4 of S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).
Pi^3 = 720*S(3,1) - 900*S(3,2) + 180*S(3,4).
zeta(3) = 28*S(3,1) - 37*S(3,2) + 7*S(3,4).
Equals Sum_{k>=1} sigma_3(k)/(k^3*exp(4*Pi*k)). - Amiram Eldar, Jun 05 2023
EXAMPLE
0.000003487356038004276054514730322548976264651146827033884525679...
MATHEMATICA
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
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Mar 02 2015
STATUS
approved