A075700 Decimal expansion of -zeta'(0).

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

Offset: 0

Author

Benoit Cloitre, Oct 02 2002

Keywords

• cons
• nonn

Comments

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

Examples

			0.91893853320467274178032...
		

Links

• 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.

Crossrefs

Cf. A019727, A061444, A257549.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/2; // G. C. Greubel, Oct 07 2018

Maple

evalf(log(2*Pi)/2,120); # Muniru A Asiru, Oct 08 2018

Mathematica

Log[Sqrt[2*Pi]] // RealDigits[#, 10, 104] & // First (* Jean-François Alcover, Apr 29 2013 *)

PARI

-zeta'(0) \\ Charles R Greathouse IV, Mar 28 2012

Formula

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

Extensions

Normalized representation (leading zero and offset) R. J. Mathar, Jan 25 2009