|
%I #13 Mar 26 2019 11:46:21
%S 0,9,1,0,8,0,3,8,1,2,4,2,5,2,1,1,1,4,5,7,1,3,5,6,0,8,8,5,5,5,8,6,7,2,
%T 6,9,2,5,0,9,6,5,8,9,2,8,4,4,4,6,3,3,0,1,5,8,8,4,1,4,9,0,2,3,1,2,1,4,
%U 5,1,0,6,3,6,0,4,2,7,5,5,5,9,9,4,1,6,3,9,8,5,5,9,9,9,3,7,5,7,4,8,8,6,7,6,8,1
%N Decimal expansion of zeta'(-1, 5/6) (negated).
%H J. Miller and V. Adamchik, <a href="https://doi.org/10.1016/S0377-0427(98)00193-9">Derivatives of the Hurwitz Zeta Function for Rational Arguments</a>, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>, formula 25.
%F Equals Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 - PolyGamma(1, 1/3)/(8*sqrt(3)*Pi) + Zeta'(-1)/6.
%F A324997 + A324998 = log(2)/36 + log(3)/72 + Zeta'(-1)/3.
%e -0.09108038124252111457135608855586726925096589284446330158841490231214...
%p evalf(Zeta(1,-1,5/6), 120);
%p evalf(Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 - Psi(1, 1/3)/(8*sqrt(3)*Pi) + Zeta(1,-1)/6, 120);
%t RealDigits[Derivative[1, 0][Zeta][-1, 5/6], 10, 120][[1]]
%t N[With[{k=1}, ((9^k-1)*(2^(2*k-1) + 1)*BernoulliB[2*k]*Pi/(8*Sqrt[3]*k*6^(2*k-1)) + BernoulliB[2*k]*(3^(2*k-1) - 1) * Log[2]/(4*k*6^(2*k-1)) + BernoulliB[2*k]*(2^(2*k-1)-1) * Log[3]/(4*k*6^(2*k-1)) + (-1)^k*(2^(2*k-1) + 1) * PolyGamma[2*k-1,1/3] / (2*Sqrt[3]*(12*Pi)^(2*k-1)) + (2^(2*k - 1) - 1)*(3^(2*k - 1) - 1)*Zeta'[1-2*k]/2/6^(2*k-1))], 120]
%o (PARI) zetahurwitz'(-1, 5/6) \\ _Michel Marcus_, Mar 24 2019
%Y Cf. A084448, A240966, A324997.
%K nonn,cons
%O 0,2
%A _Vaclav Kotesovec_, Mar 23 2019
|
