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%I #64 Feb 28 2022 12:20:21
%S 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,
%T 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,
%U 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
%N Decimal expansion of -zeta'(2) (the first derivative of the zeta function at 2).
%C 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
%D C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359.
%H G. C. Greubel, <a href="/A073002/b073002.txt">Table of n, a(n) for n = 0..10000</a>
%H D. Huylebrouck, <a href="https://lirias.kuleuven.be/bitstream/123456789/439440/1/GeneralisingWallis.pdf">Generalizing Wallis' formula</a>, American Mathematical Monthly, to appear, 2015;
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap17.html">Zeta(1,2) the derivative of Zeta function at 2</a>
%H J. B. Rosser and L. Schoenfeld, <a href="https://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>, Ill. J. Math. 6 (1) (1962) 64-94, Table IV
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html">MathWorld: Riemann Zeta Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>
%F Sum_{n >= 1} log(n) / n^2. - _N. J. A. Sloane_, Feb 19 2011
%F Pi^2(gamma + log(2Pi) - 12 log(A))/6, where A is the Glaisher-Kinkelin constant. - _Charles R Greathouse IV_, May 06 2013
%e Zeta'(2) = -0.93754825431584375370257409456786497789786028861482...
%p Zeta(1,2); evalf(%); # _R. J. Mathar_, Oct 10 2011
%t (* 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 *)
%t (* 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 *)
%o (PARI) -zeta'(2) \\ _Charles R Greathouse IV_, Mar 28 2012
%Y Cf. A201994 (2nd derivative), A201995 (3rd derivative), A252898.
%Y Cf. A244115, A261506.
%K cons,nonn
%O 0,1
%A _Robert G. Wilson v_, Aug 03 2002
%E Definition corrected by _N. J. A. Sloane_, Feb 19 2011
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