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A075700
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Decimal expansion of -zeta'(0).
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37
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9, 1, 8, 9, 3, 8, 5, 3, 3, 2, 0, 4, 6, 7, 2, 7, 4, 1, 7, 8, 0, 3, 2, 9, 7, 3, 6, 4, 0, 5, 6, 1, 7, 6, 3, 9, 8, 6, 1, 3, 9, 7, 4, 7, 3, 6, 3, 7, 7, 8, 3, 4, 1, 2, 8, 1, 7, 1, 5, 1, 5, 4, 0, 4, 8, 2, 7, 6, 5, 6, 9, 5, 9, 2, 7, 2, 6, 0, 3, 9, 7, 6, 9, 4, 7, 4, 3, 2, 9, 8, 6, 3, 5, 9, 5, 4, 1, 9, 7, 6, 2, 2, 0, 0
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OFFSET
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0,1
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COMMENTS
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The probability density function for the standard normal distribution is e^(-x^2/2 + zeta'(0)). - Rick L. Shepherd, Mar 08 2014
For every x > 0, PolyGamma(-2, x+1) - (PolyGamma(-2, x) + x*log(x) - x) equals this constant -zeta'(0), where polygamma functions of negative indices are defined for x > 0 as: PolyGamma(-1, x) = log(Gamma(x)), PolyGamma(-(n+1), x) = Integral_{t=0..x} PolyGamma(-n, x) dx, n >= 1. - Jianing Song, Apr 20 2021
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula.
Eric Weisstein's World of Mathematics, Log Gamma Function.
Eric Weisstein's World of Mathematics, Stirling's Approximation.
Wikipedia, Gamma function.
Wikipedia, Normal curve
Index entries for zeta function.
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FORMULA
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Equals log(2*Pi)/2 = A061444/2 = log(A019727).
Equals Integral_{x=0..1} log(Gamma(x)) dx. - Jean-François Alcover, Apr 29 2013
More generally, equals t-t*log(t)+Integral_{x=t..(t+1)} (log(Gamma(x)) dx for any t>=0 (the Raabe formula). - Stanislav Sykora, May 14 2015
Equals lim_{k->oo} log(k!) + k - (k + 1/2)*log(k) (by Stirling's formula). - Amiram Eldar, Aug 21 2020
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EXAMPLE
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0.91893853320467274178032...
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MAPLE
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evalf(log(2*Pi)/2, 120); # Muniru A Asiru, Oct 08 2018
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MATHEMATICA
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Log[Sqrt[2*Pi]] // RealDigits[#, 10, 104] & // First (* Jean-François Alcover, Apr 29 2013 *)
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PROG
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(PARI) -zeta'(0) \\ Charles R Greathouse IV, Mar 28 2012
(MAGMA) SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/2; // G. C. Greubel, Oct 07 2018
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CROSSREFS
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Cf. A019727, A061444, A257549.
Sequence in context: A299622 A163899 A198758 * A021843 A231931 A175615
Adjacent sequences: A075697 A075698 A075699 * A075701 A075702 A075703
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KEYWORD
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cons,nonn
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AUTHOR
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Benoit Cloitre, Oct 02 2002
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EXTENSIONS
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Normalized representation (leading zero and offset) R. J. Mathar, Jan 25 2009
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STATUS
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approved
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