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A073002 Decimal expansion of -zeta'(2) (the first derivative of the zeta function at 2). 13
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

Table of n, a(n) for n=0..104.

D. Huylebrouck, Generalizing Wallis' formula, American Mathematical Monthly, to appear, 2015;

Simon Plouffe, Zeta(1,2) the derivative of Zeta function at 2

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-Fran├žois Alcover, Apr 11 2013 *)

PROG

(PARI) -zeta'(2) \\ Charles R Greathouse IV, Mar 28, 2012

CROSSREFS

Cf. A201994 (2nd derivative), A201995 (3rd derivative), A252898.

Sequence in context: A011229 A068353 A136251 * A197836 A011282 A196823

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 August 1 14:19 EDT 2015. Contains 260182 sequences.