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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
D. Huylebrouck, Generalizing Wallis' formula, American Mathematical Monthly, to appear, 2015;
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.
Sequence in context: A068353 A346989 A136251 * A357044 A348302 A197836
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.)