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 A073002 Decimal expansion of -zeta'(2) (the first derivative of the zeta function at 2). 55
 9, 3, 7, 5, 4, 8, 2, 5, 4, 3, 1, 5, 8, 4, 3, 7, 5, 3, 7, 0, 2, 5, 7, 4, 0, 9, 4, 5, 6, 7, 8, 6, 4, 9, 7, 7, 8, 9, 7, 8, 6, 0, 2, 8, 8, 6, 1, 4, 8, 2, 9, 9, 2, 5, 8, 8, 5, 4, 3, 3, 4, 8, 0, 3, 6, 0, 4, 4, 3, 8, 1, 1, 3, 1, 2, 7, 0, 7, 5, 2, 2, 7, 9, 3, 6, 8, 9, 4, 1, 5, 1, 4, 1, 1, 5, 1, 5, 1, 7, 4, 9, 3, 1, 1, 3 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Successive derivatives of the Zeta function evaluated at x=2 round to (-1)^n * n!, for the n-th derivative, and converge with increasing n. For example, in Mathematica, Derivative[5][Zeta][2] = -120.000824333. A direct formula for the n-th derivative of Zeta at x=2 is: (-1)^n*Sum_{k>=1} log(k)^n/k^2. See also A201994 and A201995. The values of successive derivatives of Zeta(x) as x->1 are given by A252898, and are also related to the factorials. - Richard R. Forberg, Dec 30 2014 REFERENCES C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359. LINKS G. C. Greubel, Table of n, a(n) for n = 0..10000 D. Huylebrouck, Generalizing Wallis' formula, American Mathematical Monthly, to appear, 2015; Simon Plouffe, Zeta(1,2) the derivative of Zeta function at 2 J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant FORMULA Sum_{n >= 1} log(n) / n^2. - N. J. A. Sloane, Feb 19 2011 Pi^2(gamma + log(2Pi) - 12 log(A))/6, where A is the Glaisher-Kinkelin constant. - Charles R Greathouse IV, May 06 2013 EXAMPLE Zeta'(2) = -0.93754825431584375370257409456786497789786028861482... MAPLE Zeta(1, 2); evalf(%); # R. J. Mathar, Oct 10 2011 MATHEMATICA (* first do *) Needs["NumericalMath`NLimit`"], (* then *) RealDigits[ N[ ND[ Zeta[z], z, 2, WorkingPrecision -> 200, Scale -> 10^-20, Terms -> 20], 111]][[1]] (* Eric W. Weisstein, May 20 2004 *) (* from version 6 on *) RealDigits[-Zeta'[2], 10, 105] // First (* or *) RealDigits[-Pi^2/6*(EulerGamma - 12*Log[Glaisher] + Log[2*Pi]), 10, 105] // First (* Jean-François Alcover, Apr 11 2013 *) PROG (PARI) -zeta'(2) \\ Charles R Greathouse IV, Mar 28 2012 CROSSREFS Cf. A201994 (2nd derivative), A201995 (3rd derivative), A252898. Cf. A244115, A261506. Sequence in context: A068353 A346989 A136251 * A357044 A348302 A197836 Adjacent sequences: A072999 A073000 A073001 * A073003 A073004 A073005 KEYWORD cons,nonn AUTHOR Robert G. Wilson v, Aug 03 2002 EXTENSIONS Definition corrected by N. J. A. Sloane, Feb 19 2011 STATUS approved

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Last modified July 20 15:46 EDT 2024. Contains 374459 sequences. (Running on oeis4.)