login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A256591 Decimal expansion of Xi''(1/2) = 0.02297..., the second derivative of the Riemann Xi function at 1/2. 0
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 (list; constant; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..102.

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

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)).

Sequence in context: A298597 A066320 A005168 * A011149 A212990 A220265

Adjacent sequences:  A256588 A256589 A256590 * A256592 A256593 A256594

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Apr 03 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 15 08:58 EDT 2021. Contains 342977 sequences. (Running on oeis4.)