A084448 Decimal expansion of (negative of) Kinkelin constant.
1, 6, 5, 4, 2, 1, 1, 4, 3, 7, 0, 0, 4, 5, 0, 9, 2, 9, 2, 1, 3, 9, 1, 9, 6, 6, 0, 2, 4, 2, 7, 8, 0, 6, 4, 2, 7, 6, 4, 0, 3, 6, 3, 8, 0, 3, 3, 5, 2, 0, 1, 7, 8, 3, 6, 6, 6, 5, 2, 2, 3, 0, 6, 3, 5, 7, 3, 5, 9, 6, 9, 9, 6, 6, 6, 5, 7, 7, 1, 7, 2, 7, 5, 9, 5, 2, 5, 1, 0, 0, 3, 3, 2, 5, 0, 8, 7, 5, 5
Offset: 0
Examples
-0.1654211437004509292139196602427806427640363803352017836665223...
Links
- Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., Vol. 7, No. 4 (1998), pp. 343-359.
- Hermann Kinkelin, Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math., Vol. 57 (1860), pp. 122-158; alternative link. See eq. (22), p. 133.
- E. M. Wright, Asymptotic partition formulae, I: Plane partitions, Quart. J. Math., Vol. 2 (1931), pp. 177-189.
Programs
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Maple
Digits := 200; evalf(Zeta(1,-1));
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Mathematica
RealDigits[1/12 - Log[Glaisher], 10, 99] // First (* Jean-François Alcover, Feb 15 2013 *)
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PARI
-zeta'(-1) \\ Charles R Greathouse IV, Dec 12 2013
Formula
Zeta(1, -1). Almkvist gives many formulas.
Equals (1 - gamma - log(2*Pi))/12 + Zeta'(2)/(2*Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 25 2015
From Amiram Eldar, Jun 16 2021: (Start)
Equals 1/24 - gamma/3 - Sum_{k>=1} (zeta(2*k+1)-1)/((2*k+1)*(2*k+3)) = 1/12 - log(A), where A is the Glaisher-Kinkelin constant (A074962) (Kinkelin, 1860).
Equals 2 * Integral_{x>=0} x*log(x)/(exp(2*Pi*x)-1) dx = 2*A261819. (Wright, 1931). (End)
Comments