A324996 - OEIS

%I #14 Mar 26 2019 11:45:38

%S 0,5,2,2,1,2,5,8,3,0,4,5,1,4,8,3,2,8,8,8,2,8,5,6,1,5,6,5,8,6,7,5,1,1,

%T 4,9,3,7,0,5,1,1,9,9,0,9,5,4,5,5,9,0,9,4,6,0,6,9,3,5,1,0,3,9,8,3,2,6,

%U 6,9,4,6,7,6,1,7,8,7,5,6,8,8,3,6,7,1,6,0,6,8,5,3,4,2,1,9,9,2,0,2,8,4,9,4,6,0

%N Decimal expansion of zeta'(-1, 3/4) (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 24.

%F Equals Pi/32 - PolyGamma(1, 1/4)/(32*Pi) - Zeta'(-1)/8.

%F A324995 + A324996 = -Zeta'(-1)/4.

%e -0.05221258304514832888285615658675114937051199095455909460693510398326...

%p evalf(Zeta(1,-1,3/4), 120);

%p evalf(Pi/32 - Psi(1, 1/4)/(32*Pi) - Zeta(1,-1)/8, 120);

%t RealDigits[Derivative[1, 0][Zeta][-1, 3/4], 10, 120][[1]]

%t N[With[{k=1}, (4^k-1) * BernoulliB[2*k] * Pi/4^(k+1)/k + (4^(k-1) - 1) * BernoulliB[2*k] * Log[2]/k/2^(4*k-1) + (-1)^k*PolyGamma[2*k-1,1/4] / 4 / (8*Pi)^(2*k-1) - (4^k-2)*Zeta'[1-2*k]/2^(4*k)], 120]

%o (PARI) zetahurwitz'(-1, 3/4) \\ _Michel Marcus_, Mar 24 2019

%Y Cf. A084448, A240966, A324995.

%K nonn,cons

%O 0,2

%A _Vaclav Kotesovec_, Mar 23 2019