A225746 Decimal expansion of the logarithm of Glaisher's constant.
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, 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, 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
Offset: 1
Examples
0.248754477033784262547252993576113976097369713668535116999855639690693032999...
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.
Programs
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Mathematica
RealDigits[Log[Glaisher], 10, 100] // First
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PARI
1/12-zeta'(-1) \\ Charles R Greathouse IV, Dec 12 2013
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)