A114720 -id:A114720 - 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))).

A365281 Decimal expansion of the least real solution x > 0 of Gamma(1/4 + x/2)/(Pi^x*Gamma(1/4 - x/2)) = 1.

Original entry on oeis.org

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

Offset: 2

Thomas Scheuerle, Aug 31 2023

			18.56775084706966207279145836562344730...

Cf. A059750, A114720, A257870.

FindRoot[-1 + Gamma[1/4 - x/2]/(Pi^(-x) Gamma[1/4 + x/2]) == 0, {x, 18.5569, 18.5739}, WorkingPrecision -> 100]

Let x be this constant:
Gamma(1/4 - x/2)/(Pi^x*Gamma(1/4 + x/2)) = 1.
zeta((1/2) + x) = zeta((1/2) - x), where zeta is the Riemann zeta function.
(2*Pi)^(-1/2 - x)*(2*x - 1)*cos(Pi/4 + (Pi*x)/2)*Gamma(x - 1/2) = 1.
2^(1/2 - x)*Pi^(-1/2 - x)*sin(Pi/2 - (Pi*x)/2)*Gamma(1/2 + x) = 1.
Z(i*x) = -zeta((1/2) + x) = -zeta((1/2) - x), where Z is the Riemann-Siegel Z function and i is the imaginary unit. From this follows that theta(i*-x) = theta(i*x) is an odd multiple of Pi where theta is the Riemann-Siegel theta function. This can also be seen if we consider Hardy's definition of the Z function: Z(s) = Pi^(-i*s/2)*zeta((1/2) + i*s)*Gamma((1/4)+(i*s/2))^(1/2)/Gamma((1/4) - (i*s/2))^(1/2).

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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