OFFSET
0,4
REFERENCES
Bruce C. Berndt, Ramanujan Notebook part II, Infinite series, Springer Verlag, 1989, pp. 280-281.
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
Equals log(4/Pi)/4 - Pi/12 + log(Gamma(3/4)).
From Jean-François Alcover, Mar 02 2015: (Start)
This is the case k=1, m=2 of the Plouffe sum S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).
Pi = 72*S(1,1) - 96*S(1,2) + 24*S(1,4). (End)
Equals Sum_{k>=1} sigma(k)/(k*exp(2*Pi*k)). - Amiram Eldar, Jun 05 2023
EXAMPLE
0.00187268244976854611563857947996139886916289565261...
MATHEMATICA
digits = 104; S[1, 2] = NSum[1/(n*(Exp[2*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[1, 2], 10, digits] // First (* Jean-François Alcover, Mar 02 2015 *)
Join[{0, 0}, RealDigits[Log[4/Pi]/4 - Pi/12 + Log[Gamma[3/4]], 10, 100][[1]]] (* Amiram Eldar, May 21 2022 *)
PROG
(PARI) 1/4*log(4/Pi)-Pi/12+log(gamma(3/4))
CROSSREFS
KEYWORD
AUTHOR
Benoit Cloitre, Jun 21 2003
STATUS
approved