OFFSET
0,2
COMMENTS
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.
REFERENCES
H. M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0) pp. 16-18
LINKS
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
FORMULA
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))).
EXAMPLE
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...
MATHEMATICA
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]]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Apr 03 2015
STATUS
approved