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

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

Offset: 0

Author

Robert G. Wilson v, Aug 03 2002

Keywords

• cons
• nonn

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

Examples

			Zeta'(2) = -0.93754825431584375370257409456786497789786028861482...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.18, p. 157.
• 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.

Crossrefs

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

Cf. A244115, A261506.

Programs

Maple

Zeta(1,2); evalf(%, 120); # 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 *)

PARI

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

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

Extensions

Definition corrected by N. J. A. Sloane, Feb 19 2011