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
KEYWORD
cons,nonn
AUTHOR
Robert G. Wilson v, Aug 03 2002
EXTENSIONS
Definition corrected by N. J. A. Sloane, Feb 19 2011
STATUS
approved