Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T03:23:38.935Z Has data issue: false hasContentIssue false
  • English
  • Français

Riemann-Siegel sums via stationary phase

Published online by Cambridge University Press:  17 April 2009

E. O. Tuck
E. O. Tuck
Affiliation:
Applied Mathematics Department, The University of Adelaide, Adelaide SA 5005, Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new representation is obtained for the Riemann ξ 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 2 , October 2005 , pp. 325 - 328
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Abramowitz, M. and Stegun, I.A., Handbook of mathematical functions with formulas, graphs and mathematical tables (Dover Publications, New York, 1964).Google Scholar
[2]Berry, M.V. and Keating, J.P., ‘A new asymptotic representation for ζ(½ + it) and quantum spectral determinants’, Proc. Roy. Soc. Lond. Ser. A 437 (1992), 151–173.Google Scholar
[3]Borwein, J.M., Bradley, D.M. and Crandall, P.E., ‘Computational strategies for the Riemann zeta function’, J. Comput. Appl. Math. 121 (2000), 247–296.CrossRefGoogle Scholar
[4]Edwards, H.M., Riemann's zeta function (Academic Press, New York, 1974).Google Scholar
[5]Hill, J.M., ‘On some integrals involving functions ɸ(x) such that ’, J. Math. Anal. Appl. 309 (2005), 256–270.CrossRefGoogle Scholar
[6]Olver, F.W.J., Introduction to asymptotics and special functions (Academic Press, New York, London, 1974).Google Scholar
[7]Stoker, J.J., Water waves (Interscience Publishers Inc., New York, 1957).Google Scholar
[8]Titchmarsh, E.C., The theory of the Riemann zeta-function, (2nd edition) (Oxford University Press, New York, 1986).Google Scholar
[9]Tuck, E.O., Collins, J.I. and Wells, W., ‘On ship wave patterns and their spectra’, J. Ship Res. 15 (1971), 11–21.CrossRefGoogle Scholar