OFFSET
1,2
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 135.
LINKS
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, The Ramanujan Journal, Vol. 16, No. 3 (2008), pp. 247-270; arXiv preprint, arXiv:math/0506319 [math.NT], 2005-2006.
Jean-Christophe Pain, Two integral representations for the logarithm of the Glaisher-Kinkelin constant, arXiv:2405.05264 [math.GM], 2024.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.
FORMULA
Equals 1/12 - zeta'(-1).
Also equals (gamma + log(2*Pi))/12 -zeta'(2)/(2*Pi^2).
From Amiram Eldar, Apr 15 2021: (Start)
Equals lim_{n->oo} (Sum_{k=1..n} k*log(k) - (n^2/2 + n/2 + 1/12)*log(n) + n^2/4).
Equals 1/8 + (1/2) * Sum_{n>=0} ((1/(n+1)) * Sum_{k=0..n} (-1)^(k+1) * binomial(n,k) * (k+1)^2 * log(k+1)) (Guillera and Sondow, 2008). (End)
EXAMPLE
0.248754477033784262547252993576113976097369713668535116999855639690693032999...
MATHEMATICA
RealDigits[Log[Glaisher], 10, 100] // First
PROG
(PARI) 1/12-zeta'(-1) \\ Charles R Greathouse IV, Dec 12 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, May 14 2013
STATUS
approved