A252244 -id:A252244 - cp's OEIS frontend

A252245 Decimal expansion of zeta'''(1/2) (negated).

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

Offset: 2

Jean-François Alcover, Dec 16 2014

			-96.003309245319070097389767220695459302514018846555728...

MathOverflow, Zeta(3) in terms of derivatives of zeta at 1/2 and pi
Eric Weisstein's MathWorld, Riemann Zeta Function

Cf. A002117, A059750, A114875, A252244.

RealDigits[Zeta'''[1/2], 10, 104] // First

zeta(3) = (1/7)*(-Pi^3/4 + (2*zeta'(1/2)^3 - 3*zeta(1/2)*zeta'(1/2)*zeta''(1/2) + zeta(1/2)^2*zeta'''(1/2))/zeta(1/2)^3).

A256591 Decimal expansion of Xi''(1/2) = 0.02297..., the second derivative of the Riemann Xi function at 1/2.

Original entry on oeis.org

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, 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, 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

Offset: 0

Jean-François Alcover, Apr 03 2015

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.

			0.022971944315145437535249876497632170264593013837589...
Are also listed in the Borwein paper the Xi derivatives of order 4 and 6:
Xi^(4)(1/2) = 0.002962848433687632165368...
Xi^(6)(1/2) = 0.000599295946597579491843...

H. M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0) pp. 16-18

Jonathan M. Borwein, David M. Bradley, Richard E. Crandall, Computational strategies for the Riemann zeta function, Journal of Computational and Applied Mathematics 121 (2000) p. 289.
Eric Weisstein's MathWorld, Xi-Function
Wikipedia, Riemann Xi function

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)).

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]]]

Xi(s) = 1/2*s*(s-1)*Pi^(-s/2)*Gamma(s/2)*zeta(s).

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))).

cp's OEIS Frontend

A252245 Decimal expansion of zeta'''(1/2) (negated).

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