Bull. Austral. Math. Soc. 72(2) pp.325--328, 2005.
Riemann-Siegel sums via stationary phase
E.O. Tuck |
I thank Jim Hill for discussions of this topic.
Abstract
A new representation is obtained for the
Riemann 
function,
in the form of a series of integrals, multiplied by an exponential
factor capturing the correct decay rate for large imaginary
argument. Each term in this series then has a simple
stationary-phase asymptote, the total agreeing with the
Riemann-Siegel sum.
function,
in the form of a series of integrals, multiplied by an exponential
factor capturing the correct decay rate for large imaginary
argument. Each term in this series then has a simple
stationary-phase asymptote, the total agreeing with the
Riemann-Siegel sum.Click to download PDF of this article (free access until July 2006)
or get the no-frills version
[an error occurred while processing this directive]| (Metadata: XML, RSS, BibTeX) | MathSciNet: MR2183413 | Z'blatt-MATH: 02246394 |
References
- M. Abramowitz and I.A. Stegun;
Handbook of mathematical functions with formulas, graphs and mathematical tables (Dover Publications, New York, 1964). MR1225604 - M.V. Berry and J.P. Keating;
A new asymptotic representation for ζ(1/2+it) and quantum spectral determinants,
Proc. Roy. Soc. Lond. Ser. A 437 (1992), pp. 151--173. MR1177749 - J.M. Borwein, D.M. Bradley and P.E. Crandall;
Computational strategies for the Riemann zeta function,
J. Comput. Appl. Math. 121 (2000), pp. 247--296. MR1780051 - H.M. Edwards;
Riemann's zeta function (Academic Press, New York, 1974). MR0466039, MR1854455 - J.M. Hill;
On some integrals involving functions φ(x) such that φ(1/x)=√{x}φ(x),
J. Math. Anal. Appl. 309 (2005), pp. 256--270. MR2154040 - F.W.J. Olver;
Introduction to asymptotics and special functions (Academic Press, New York, London, 1974). MR435697 - J.J. Stoker;
Water waves (Interscience Publishers Inc., New York, 1957). MR103672 - E.C. Titchmarsh;
The theory of the Riemann zeta-function,
(2nd edition) (Oxford University Press, New York, 1986). MR882550 - E.O. Tuck, J.I. Collins and W. Wells;
On ship wave patterns and their spectra,
J. Ship Res. 15 (1971), pp. 11--21.
ISSN 0004-9727