A282899 -id:A282899 - cp's OEIS frontend

A282898 Numerator of the coefficients of the series expansion of the Riemann-Siegel theta function at infinity.

1, 7, 31, 127, 511, 1414477, 8191, 118518239, 5749691557, 91546277357, 23273283019, 1982765468311237, 22076500342261, 455371239541065869, 925118910976041358111, 16555640865486520478399, 1302480594081611886641, 904185845619475242495834469891

Offset: 1

Mats Granvik and Robert G. Wilson v, Feb 24 2017

See "RiemannSiegelTheta" in the help file of Mathematica, Series expansion at infinity.

Seiichi Manyama, Table of n, a(n) for n = 1..275
Richard P. Brent, On asymptotic approximations to the log-Gamma and Riemann-Siegel theta functions, arXiv:1609.03682 [math.NA], 2016.
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions
Wikipedia, Riemann-Siegel theta function
Wolfram Language and System, RiemannSiegelTheta

Cf. A114721, A282899.

Differs from A036282.

Numerator[ DeleteCases[ CoefficientList[ CoefficientList[ Series[ RiemannSiegelTheta[ t], {t, Infinity, 41}], 1/t^_] + Pi/8 + t/2 + t*Log[2]/2 + t*Log[Pi]/2 + t*Log[1/t]/2, 1/t][[1]], 0]]

A114721 Denominator of expansion of RiemannSiegelTheta(t) about infinity.

Original entry on oeis.org

48, 5760, 80640, 430080, 1216512, 1476034560, 2555904, 8021606400, 64012419072, 131491430400, 3472883712, 25282593423360, 20132659200, 25222195445760, 2675794690179072, 2172909854392320, 6803228196864

Offset: 1

Eric W. Weisstein, Dec 27 2005

			RiemannSiegelTheta(t) = -Pi/8 + t*(-1/2 - log(2)/2 - log(Pi)/2 - log(t^(-1))/2) + 1/(48*t) + 7/(5760*t^3) + 31/(80640*t^5) + ...

H. M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0), p. 120.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
Simon Plouffe, On the values of the functions zeta and gamma, arXiv preprint arXiv:1310.7195, 2013.
Eric Weisstein's World of Mathematics, Riemann-Siegel Function

Cf. A036282, A282898 (numerators), A282899.

a[n_] := (-1)^n*BernoulliB[2*n, 1/2]/(4*n*(2*n-1)) // Denominator; Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Aug 04 2014 *)

a(n) = denominator(subst(bernpol(2*n), x, 1/2)/(4*n*(2*n-1))); \\ Michel Marcus, Jun 20 2018

a(n) is the denominator of (-1)^n*BernoulliB(2*n, 1/2)/(4*n*(2*n-1)).

cp's OEIS Frontend

A282898 Numerator of the coefficients of the series expansion of the Riemann-Siegel theta function at infinity.

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A114721 Denominator of expansion of RiemannSiegelTheta(t) about infinity.

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