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%I #8 Sep 29 2018 18:48:08
%S 0,2,2,9,7,1,9,4,4,3,1,5,1,4,5,4,3,7,5,3,5,2,4,9,8,7,6,4,9,7,6,3,2,1,
%T 7,0,2,6,4,5,9,3,0,1,3,8,3,7,5,8,9,0,6,3,4,9,9,1,4,4,6,2,2,1,6,5,1,8,
%U 3,6,3,1,8,5,8,8,9,2,5,5,3,8,0,9,6,7,0,2,2,7,6,7,1,2,1,4,1,7,8,0,3,2,3
%N Decimal expansion of Xi''(1/2) = 0.02297..., the second derivative of the Riemann Xi function at 1/2.
%C As mentioned in the paper by Borwein et al., the Riemann hypothesis is equivalent to a positivity condition on every even-order derivative of the Xi function at the point s = 1/2.
%D H. M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0) pp. 16-18
%H Jonathan M. Borwein, David M. Bradley, Richard E. Crandall, <a href="http://dx.doi.org/10.1016/S0377-0427(00)00336-8">Computational strategies for the Riemann zeta function</a>, Journal of Computational and Applied Mathematics 121 (2000) p. 289.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Xi-Function.html">Xi-Function</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_Xi_function">Riemann Xi function</a>
%F Xi(s) = 1/2*s*(s-1)*Pi^(-s/2)*Gamma(s/2)*zeta(s).
%F Xi''(1/2) = (-(32*Pi^(1/4))^(-1))*Gamma(1/4)*((-32 + (log(Pi) - PolyGamma(1/4))^2 + PolyGamma(1, 1/4))*zeta(1/2) + 4*((-log(Pi) + PolyGamma(1/4))*zeta'(1/2) + zeta''(1/2))).
%e 0.022971944315145437535249876497632170264593013837589...
%e Are also listed in the Borwein paper the Xi derivatives of order 4 and 6:
%e Xi^(4)(1/2) = 0.002962848433687632165368...
%e Xi^(6)(1/2) = 0.000599295946597579491843...
%t d2 = (-(32*Pi^(1/4))^(-1))*Gamma[1/4]*((-32 + (Log[Pi] - PolyGamma[1/4])^2 + PolyGamma[1, 1/4])*Zeta[1/2] + 4*((-Log[Pi] + PolyGamma[1/4])*Zeta'[1/2] + Zeta''[1/2])); Join[{0}, First[RealDigits[d2, 10, 102]]]
%Y Cf. A020777 (PolyGamma(1/4)), A059750 (zeta(1/2)), A068466 (Gamma(1/4)), A114720 (Xi(1/2)), A114875 (zeta'(1/2)), A252244 (zeta''(1/2)).
%K nonn,cons
%O 0,2
%A _Jean-François Alcover_, Apr 03 2015
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