OFFSET
1,3
COMMENTS
Also known as the second Bendersky constant.
This is likely the same as the constant B considered in section 3 of the Choi and Srivastava link. - R. J. Mathar, Oct 03 2016
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..2002
J. Choi and H. M. Srivastava, Certain classes of series involving the zeta function, J. Math. Annal. Applic. 231 (1999) 91-117.
K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.
Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant
FORMULA
A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(0) = sqrt(2*Pi) (A019727),
A(1) = A = Glaisher-Kinkelin constant (A074962),
A(2) = exp(-zeta'(-2)) = exp(zeta(3)/(4*Pi^2)).
Equals exp(-A240966). - Vaclav Kotesovec, Feb 22 2015
EXAMPLE
1.03091675219739211419331309646694229...
MATHEMATICA
RealDigits[Exp[Zeta[3]/(4*Pi^2)], 10, 99] // First
RealDigits[Exp[N[(BernoulliB[2]/4)*(Zeta[3]/Zeta[2]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)
PROG
(PARI) exp(zeta(3)/(4*Pi^2)) \\ Felix Fröhlich, Jun 27 2019
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 02 2014
STATUS
approved