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Prog. Theor. Phys. Supplement No.166 (2007) pp. 19-36

[ Full Text PDF : FREE ACCESS (514K) ]

Riemann Zeta Function and Quantum Chaos

Eugène Bogomolny

CNRS, Université Paris-Sud, UMR 8626,
Laboratoire de Physique Théorique et Modèles Statistiques,
91405 Orsay, France

Abstract:

A brief review of recent developments in the theory of the Riemann zeta function inspired by ideas and methods of quantum chaos is given.


URL : http://ptp.ipap.jp/link?PTPS/166/19/
DOI : 10.1143/PTPS.166.19

[ Full Text PDF : FREE ACCESS (514K) ] Citation:


References:

  1. M. V. Berry, in Lecture Notes in Phys. 263, ed. T. H. Seligman and H. Nishioka (Springer-Verlag, New York, 1986), p. 1.
  2. M. V. Berry and J. P. Keating, SIAM Review 41 (1999), 236.
  3. D. A. Hejhal, Duke Math. J. 43 (1976), 441.
  4. O. Bohigas, M. J. Giannoni and C. Schmit, Phys. Rev. Lett. 52 (1984), 1[APS].
  5. H. L. Montgomery, Proc. of Symposia in Pure Mathematics 24 (1973), 181.
  6. G. H. Hardy and J. E. Littlewood, Acta Math. 44 (1923), 1.
  7. E. B. Bogomolny and J. P. Keating, Phys. Rev. Lett. 77 (1996), 1472[APS].
  8. E. Bogomolny, O. Bohigas, P. Leboeuf and A. G. Monastra, J. of Phys. A 39 (2006), 10743[IoP STACKS].
  9. J. P. Keating and N. Snaith, Commun. Math. Phys. 214 (2000), 57[CrossRef].
  10. J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubenstein and N. C. Snaith, Proc. R. Soc. London 400 (2005), 33.
  11. J. B. Conrey and N. C. Snaith, math.NT/0509480[e-print arXiv].
  12. H. M. Edwards, Riemann's zeta function (Academic Press, New York, London, 1974).
  13. D. A. Hejhal, v. I, Lecture Notes in Math. 548 (Sprihger, Berlin, 1976), p. 516; v. II, Lecture Notes in Math. 1001 (Sprihger, Berlin, 1983), p. 806.
  14. E. Bogomolny, “Quantum and Arithmetical Chaos”, in Frontiers in Number Theory, Physics and Geometry, Les Houches, 2003.
  15. A. M. Odlyzko, http://www.dtc.umn.edu/~odlyzko/
  16. M. C. Gutzwiller, J. Math. Phys. 12 (1971), 343[CrossRef].
  17. M. C. Gutzwiller, Chaos in classical and quantum mechanics (Springer-Verlag, New York, 1990).
  18. P. Gaspard, Chaos, scattering and statistical mechanics (Cambridge Univ. Press, Cambridge, 1998).
  19. A. Connes, CR. Acad. Sci. Paris 323 (1996), 1231.
  20. M. V. Berry and J. P. Keating, in Supersymmetry and trace formulae: chaos and disorder, ed. J. P. Keating, D. E. Khmelnitskii and I. V. Lerner (Plemium, New York, 1998), p. 355.
  21. M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, New York, 1991).
  22. A. M. Odlyzko, in Amer. Math. Soc. Contemporary Math. series, no. 290, ed. M. van Frankenhuysen and M. L. Lapidus (2001), p. 139.
  23. E. B. Bogomolny, in “New Directions in Quantum Chaos”, Proceedings of the International School of Physics “Enrico Fermi”, course CXLIII, Varenna, 2000, ed. G. Casati, I. Guarneri and U. Smilansky, p. 333.
  24. M. V. Berry, Proc. R. Soc. London A 400 (1985), 229.
  25. E. B. Bogomolny and J. P. Keating, Nonlinearity 8 (1995), 1115[IoP STACKS]; Nonlinearity 9 (1995), 911[IoP STACKS].