The holomorphic flow of the Riemann zeta function

Broughan, Kevin; Barnett, A.

doi:10.1090/S0025-5718-03-01529-1

The holomorphic flow of the Riemann zeta function
HTML articles powered by AMS MathViewer

by Kevin A. Broughan and A. Ross Barnett PDF

Math. Comp. 73 (2004), 987-1004 Request permission

Corrigendum: Math. Comp. 76 (2007), 2249-2250.

Abstract:

The flow of the Riemann zeta function, $\dot {s}=\zeta (s)$, is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica. The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.

References

Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
M. V. Berry and J. P. Keating, A new asymptotic representation for $\zeta (\frac 12+it)$ and quantum spectral determinants, Proc. Roy. Soc. London Ser. A 437 (1992), no. 1899, 151–173. MR 1177749, DOI 10.1098/rspa.1992.0053
Dunham Jackson, A class of orthogonal functions on plane curves, Ann. of Math. (2) 40 (1939), 521–532. MR 80, DOI 10.2307/1968936
Jonathan M. Borwein, David M. Bradley, and Richard E. Crandall, Computational strategies for the Riemann zeta function, J. Comput. Appl. Math. 121 (2000), no. 1-2, 247–296. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780051, DOI 10.1016/S0377-0427(00)00336-8
P. Borwein, An efficient algorithm for the Riemann zeta function, Constructive, experimental, and nonlinear analysis (Limoges, 1999) CRC Math. Model. Ser., vol. 27, CRC, Boca Raton, FL, 2000, pp. 29–34. MR 1777614
K. A. Broughan, Holomorphic flows on simply connected domains have no limit cycle, Meccanica 38(6) (2003), 699–709.
K. A. Broughan, http://www.math.waikato.ac.nz/~kab, Web site for zeta phase portraits and data for the zeros encoding.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann zeta function, Royal Society Mathematical Tables, Vol. 6, Cambridge University Press, New York, 1960. MR 0117905
William F. Galway, Computing the Riemann zeta function by numerical quadrature, Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999) Contemp. Math., vol. 290, Amer. Math. Soc., Providence, RI, 2001, pp. 81–91. MR 1868470, DOI 10.1090/conm/290/04575
P. Godfrey, An efficient algorithm for the Riemann zeta function, http://www.mathworks. com/support/ftp zeta.m, etan.m (2000)
U. D. Jentschura, P. J. Mohr, G. Soff, E. J. Weniger, Convergence acceleration via combined nonlinesr-condensation transformations, Computer Physics Comm. 116 (1999), 28-54.
J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), no. 174, 667–681. MR 829637, DOI 10.1090/S0025-5718-1986-0829637-3
A. M. Odlyzko and A. Schönhage, Fast algorithms for multiple evaluations of the Riemann zeta function, Trans. Amer. Math. Soc. 309 (1988), no. 2, 797–809. MR 961614, DOI 10.1090/S0002-9947-1988-0961614-2
Andrew M. Odlyzko, Analytic computations in number theory, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 451–463. MR 1314883, DOI 10.1090/psapm/048/1314883
A. M. Odlyzko, The first 100,000 zeros of the Riemann zeta function, http://www. research.att.com/$\sim$amo, (1997).
S. J. Patterson, An introduction to the theory of the Riemann zeta-function, Cambridge Studies in Advanced Mathematics, vol. 14, Cambridge University Press, Cambridge, 1988. MR 933558, DOI 10.1017/CBO9780511623707
Lawrence Perko, Differential equations and dynamical systems, 2nd ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 1996. MR 1418638, DOI 10.1007/978-1-4684-0249-0
A. Speiser, Geometrisches zur Riemannschen Zetafunktion, Math. Annalen 110 (1934), 514-521.
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550