Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 2 10.wxyz/CV72P2 http://www.austms.org.au/Publ/Bulletin/V72P2/ E.O. Tuck 14 February 2006 2006 2 14 325 328 722-5212-Tuck-2005 10.wxyz/C2005V72P2p325 http://www.austms.org.au/Publ/Bulletin/V72P2/722-5212-Tuck/ A new representation is obtained for the Riemann $\xi $ 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. 33E20, 41A60 MR2183413 02246394 I thank Jim Hill for discussions of this topic. M. Abramowitz and I.A. Stegun Dover Publications MR1225604 M. Abramowitz and I.A. Stegun; \textit{Handbook of mathematical functions with formulas, graphs and mathematical tables} (Dover Publications, New York, 1964). M.V. Berry and J.P. Keating Proc. Roy. Soc. Lond. Ser. A MR1177749 M.V. Berry and J.P. Keating; A new asymptotic representation for $\zeta (1/2+it)$ and quantum spectral determinants, \textit{Proc. Roy. Soc. Lond. Ser. A} \textbf{437} (1992), pp.~151--173. J.M. Borwein, D.M. Bradley and P.E. Crandall J. Comput. Appl. Math. MR1780051 J.M. Borwein, D.M. Bradley and P.E. Crandall; Computational strategies for the Riemann zeta function, \textit{J. Comput. Appl. Math.} \textbf{121} (2000), pp.~247--296. H.M. Edwards Academic Press MR0466039, MR1854455 H.M. Edwards; \textit{Riemann's zeta function} (Academic Press, New York, 1974). J.M. Hill J. Math. Anal. Appl. MR2154040 J.M. Hill; On some integrals involving functions $\phi (x)$ such that $\phi (1/x)=\sqrt {x}\phi (x)$, \textit{J. Math. Anal. Appl.} \textbf{309} (2005), pp.~256--270. F.W.J. Olver Academic Press MR435697 F.W.J. Olver; \textit{Introduction to asymptotics and special functions} (Academic Press, New York, London, 1974). J.J. Stoker Interscience Publishers Inc. MR103672 J.J. Stoker; \textit{Water waves} (Interscience Publishers Inc., New York, 1957). E.C. Titchmarsh (2nd edition) Oxford University Press MR882550 E.C. Titchmarsh; \textit{The theory of the Riemann zeta-function}, (2nd edition) (Oxford University Press, New York, 1986). E.O. Tuck, J.I. Collins and W. Wells J. Ship Res. E.O. Tuck, J.I. Collins and W. Wells; On ship wave patterns and their spectra, \textit{J. Ship Res.} \textbf{15} (1971), pp.~11--21.