A225746 - OEIS

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

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OFFSET

1,2

REFERENCES

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

LINKS

Table of n, a(n) for n=1..101.

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

Cf. A001620, A073002, A074962, A084448.

Sequence in context: A201568 A029898 A153130 * A021406 A065075 A001370

Adjacent sequences: A225743 A225744 A225745 * A225747 A225748 A225749

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, May 14 2013

STATUS

approved