A225746 - OEIS

%I #19 May 10 2024 02:43:43

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

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

%U 9,3,0,3,2,9,9,9,9,1,0,5,0,6,0,9,2,8,5,8,4,3,3,6,6,5,8,4,2,0,8,8,8

%N Decimal expansion of the logarithm of Glaisher's constant.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 135.

%H Jesús Guillera and Jonathan Sondow, <a href="https://doi.org/10.1007/s11139-007-9102-0">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, The Ramanujan Journal, Vol. 16, No. 3 (2008), pp. 247-270; <a href="https://arxiv.org/abs/math/0506319">arXiv preprint</a>, arXiv:math/0506319 [math.NT], 2005-2006.

%H Jean-Christophe Pain, <a href="https://arxiv.org/abs/2405.05264">Two integral representations for the logarithm of the Glaisher-Kinkelin constant</a>, arXiv:2405.05264 [math.GM], 2024.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>.

%F Equals 1/12 - zeta'(-1).

%F Also equals (gamma + log(2*Pi))/12 -zeta'(2)/(2*Pi^2).

%F From _Amiram Eldar_, Apr 15 2021: (Start)

%F Equals lim_{n->oo} (Sum_{k=1..n} k*log(k) - (n^2/2 + n/2 + 1/12)*log(n) + n^2/4).

%F 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)

%e 0.248754477033784262547252993576113976097369713668535116999855639690693032999...

%t RealDigits[Log[Glaisher], 10, 100] // First

%o (PARI) 1/12-zeta'(-1) \\ _Charles R Greathouse IV_, Dec 12 2013

%Y Cf. A001620, A073002, A074962, A084448.

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, May 14 2013