A020777 -id:A020777 - cp's OEIS frontend

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

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))).

A375771 Decimal expansion of the absolute value of the second derivative of the digamma function at 1/4.

Original entry on oeis.org

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

Offset: 3

R. J. Mathar, Aug 27 2024

			psi''(1/4) = -129.3277399...

Ernst D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function _{q+1}F_q, J. Comp. Appl. Math. 78 (1997) 79-95.

Cf. A020777 (psi(1/4)), A282823 (psi'(1/4)).

Maple
```
2*(Pi^3+28*Zeta(3)); evalf(%) ;
```

RealDigits[PolyGamma[2, 1/4], 10, 105][[1]] (* Vaclav Kotesovec, Aug 27 2024 *)

psi''(1/4) = -2*(Pi^3+28*zeta(3)).

cp's OEIS Frontend

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

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