

A073002


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


25



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
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OFFSET

0,1


COMMENTS

Successive derivatives of the Zeta function evaluated at x=2 round to (1)^n * n!, for the nth derivative, and converge with increasing n. For example, in Mathematica, Derivative[5][Zeta][2] = 120.000824333. A direct formula for the nth 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, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 6494, Table IV
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function
Eric Weisstein's World of Mathematics, GlaisherKinkelin 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 GlaisherKinkelin 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 (* JeanFrançois 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: 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



