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Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions


Authors: Philippe Biane, Jim Pitman and Marc Yor
Journal: Bull. Amer. Math. Soc. 38 (2001), 435-465
MSC (2000): Primary 11M06, 60J65, 60E07
DOI: https://doi.org/10.1090/S0273-0979-01-00912-0
Published electronically: June 12, 2001
MathSciNet review: 1848256
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Abstract:

This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann's zeta function which are related to these laws.


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  • 1. D.J. Aldous, The continuum random tree II: an overview, Stochastic Analysis (M.T. Barlow and N.H. Bingham, eds.), Cambridge University Press, 1991, pp. 23-70. MR 93f:60010
  • 2. K. S. Alexander, K. Baclawski, and G.-C. Rota, A stochastic interpretation of the Riemann zeta function, Proc. Nat. Acad. Sci. U.S.A. 90 (1993), no. 2, 697-699. MR 94m:11096
  • 3. L. Alili, On some hyperbolic principal values of brownian local times, Exponential functionals and principal values related to Brownian motion (M. Yor, ed.), Biblioteca de la Revista Matemática Ibero-Americana, 1997, pp. 131-154. MR 99h:60152
  • 4. L. Alili, C. Donati-Martin, and M. Yor, Une identité en loi remarquable pour l'excursion brownienne normalisée, Exponential functionals and principal values related to Brownian motion (M. Yor, ed.), Biblioteca de la Revista Matemática Ibero-Americana, 1997, pp. 155-180. MR 99i:60153
  • 5. S. Asmussen, P. Glynn, and J. Pitman, Discretization error in simulation of one-dimensional reflecting Brownian motion, Ann. Applied Prob. 5 (1995), 875-896. MR 97e:65156
  • 6. R. Bellman, A brief introduction to theta functions, Holt, Rinehart and Winston, 1961. MR 23:A2556
  • 7. M. V. Berry and J. P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Rev. 41 (1999), no. 2, 236-266 (electronic). MR 2000f:11107
  • 8. J. Bertoin and J. Pitman, Path transformations connecting Brownian bridge, excursion and meander, Bull. Sci. Math. (2) 118 (1994), 147-166. MR 95b:60097
  • 9. Ph. Biane, Decompositions of Brownian trajectories and some applications, Probability and Statistics; Rencontres Franco-Chinoises en Probabilités et Statistiques; Proceedings of the Wuhan meeting (A. Badrikian, P.-A. Meyer, and J.-A. Yan, eds.), World Scientific, 1993, pp. 51-76.
  • 10. Ph. Biane and M. Yor, Valeurs principales associées aux temps locaux Browniens, Bull. Sci. Math. (2) 111 (1987), 23-101. MR 88g:60188
  • 11. P. Billingsley, Probability and measure, Wiley, New York, 1995, 3rd ed. MR 95k:60001
  • 12. -, Convergence of probability measures, second ed., John Wiley & Sons Inc., New York, 1999, A Wiley-Interscience Publication. MR 2000e:60008
  • 13. E. Bombieri and J. C. Lagarias, Complements to Li's criterion for the Riemann hypothesis, J. Number Theory 77 (1999), no. 2, 274-287. MR 2000h:11092
  • 14. J.-B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), no. 3, 411-457. MR 96m:46112
  • 15. D. Bump, Automorphic Forms and Representations, Cambridge Univ. Press, Cambridge, 1997. MR 97k:11080
  • 16. J. T. Chang and Y. Peres, Ladder heights, Gaussian random walks, and the Riemann zeta function, Ann. Probab. 25 (1997), 787-802. MR 98c:60086
  • 17. K. L. Chung, A course in probability theory, Academic Press, 1974, 2nd ed. MR 49:11579
  • 18. -, Excursions in Brownian motion, Arkiv für Matematik 14 (1976), 155-177. MR 57:7791
  • 19. -, A cluster of great formulas, Acta Math. Acad. Sci. Hungar 39 (1982), 65-67. MR 83g:60029
  • 20. Z. Ciesielski and S. J. Taylor, First passage times and sojourn density for brownian motion in space and the exact hausdorff measure of the sample path, Trans. Amer. Math. Soc. 103 (1962), 434-450. MR 26:816
  • 21. H. Davenport, The Higher Arithmetic (Sixth edition, 1992), Cambridge Univ. Press, Cambridge, 1952. MR 14:352e; MR 93i:11001
  • 22. -, Multiplicative Number Theory, Springer-Verlag, New York, 1980, Second Edition, revised by H. L. Montgomery. MR 82m:10001
  • 23. A. Denjoy, Probabilités confirmant l'hypothèse de Riemann sur les zéros de $\zeta (s)$, C. R. Acad. Sci. Paris 259 (1964), 3143-3145. MR 31:133
  • 24. C. Donati-Martin and M. Yor, Some Brownian functionals and their laws, Ann. Probab. 25 (1997), no. 3, 1011-1058. MR 98e:60135
  • 25. J. Doob, Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Stat. 20 (1949), 393-403. MR 11:43a
  • 26. R. Durrett, Probability: Theory and examples, Duxbury Press, 1996, 2nd ed. MR 98m:60001
  • 27. R. Durrett, D. L. Iglehart, and D. R. Miller, Weak convergence to Brownian meander and Brownian excursion, Ann. Probab. 5 (1977), 117-129. MR 55:9300
  • 28. H.M. Edwards, Riemann's Zeta Function, Academic Press, New York, 1974. MR 57:5922
  • 29. L. Ehrenpreis, Fourier analysis, partial differential equations, and automorphic functions, Theta functions--Bowdoin 1987, Part 2 (Brunswick, ME, 1987), Amer. Math. Soc., Providence, RI, 1989, pp. 45-100. MR 91k:11036
  • 30. P. D. T. A. Elliott, The Riemann zeta function and coin tossing, J. Reine Angew. Math. 254 (1972), 100-109. MR 47:1761
  • 31. W. Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Stat. 22 (1951), 427-432. MR 13:140i
  • 32. I. I. Gikhman, On a nonparametric criterion of homogeneity for $k$ samples, Theory Probab. Appl. 2 (1957), 369-373.
  • 33. B. V. Gnedenko, Kriterien für die Unverändlichkeit der Wahrscheinlichkeitsverteilung von zwei unabhängigen Stichprobenreihen (in Russian), Math. Nachrichten. 12 (1954), 29-66. MR 16:498c
  • 34. S. W. Golomb, A class of probability distributions on the integers, J. Number Theory 2 (1970), 189-192.
  • 35. A. Hald, The early history of the cumulants and the Gram-Charlier series, International Statistical Review 68 (2000), 137-153.
  • 36. G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 11:25a
  • 37. Y. Hu, Z. Shi, and M. Yor, Some applications of Lévy's area formula to pseudo-Brownian and pseudo-Bessel bridges, Exponential functionals and principal values of Brownian motion, Biblioteca de la Revista Matematica Ibero-Americana, Madrid, 1996/1997. MR 99j:60122
  • 38. J. P. Imhof, On the range of Brownian motion and its inverse process, Ann. Probab. 13 (1985), 1011-1017. MR 86m:60195
  • 39. K. Itô and H. P. McKean, Diffusion processes and their sample paths, Springer, 1965. MR 33:8031
  • 40. A. Ivic, The Riemann Zeta-Function, Wiley, New York, 1985. MR 87d:11062
  • 41. P. C. Joshi and S. Chakraborty, Moments of Cauchy order statistics via Riemann zeta functions, Statistical theory and applications (H. N. Nagaraja, P. K. Sen, and D. F. Morrison, eds.), Springer, 1996, pp. 117-127. MR 98f:62149
  • 42. W. D. Kaigh, An invariance principle for random walk conditioned by a late return to zero, Ann. Probab. 4 (1976), 115 - 121. MR 54:3786
  • 43. N. M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society, Providence, RI, 1999. MR 2000b:11070
  • 44. -, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 1-26. MR 2000f:11114
  • 45. D. P. Kennedy, The distribution of the maximum Brownian excursion, J. Appl. Prob. 13 (1976), 371-376. MR 53:6769
  • 46. J. Kent, Some probabilistic properties of Bessel functions, Annals of Probability 6 (1978), 760-770. MR 58:18750
  • 47. J. T. Kent, Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 309-319. MR 81i:60072
  • 48. -, The spectral decomposition of a diffusion hitting time, Ann. Probab. 10 (1982), no. 1, 207-219. MR 83f:60103
  • 49. J. Kiefer, K-sample analogues of the Kolmogorov-Smirnov and Cramér-von Mises tests, Ann. Math. Stat. 30 (1959), 420-447. MR 21:1668
  • 50. A. Knauf, The number-theoretical spin chain and the Riemann zeroes, Comm. Math. Phys. 196 (1998), no. 3, 703-731. MR 2000d:11108
  • 51. F. B. Knight, On sojourn times of killed Brownian motion, Séminaire de Probabilités XII, Springer, 1978, Lecture Notes in Math. 649, pp. 428-445. MR 81a:60090
  • 52. -, Inverse local times, positive sojourns, and maxima for Brownian motion, Colloque Paul Lévy sur les Processus Stochastiques, Société Mathématique de France, 1988, Astérisque 157-158, pp. 233-247. MR 89m:60194
  • 53. A. N. Kolmogorov, Sulla determinazione empirica delle leggi di probabilita, Giorn. Ist. Ital. Attuari 4 (1933), 1-11.
  • 54. J. C. Lagarias and E. Rains, On a two-variable zeta function for number fields, available via http://front.math.ucdavis.edu/math.NT/0104176, 2001.
  • 55. E. Landau, Euler und die Funktionalgleichung der Riemannschen Zetafunktion, Bibliotheca Mathematica 7 (1906-1907), 69-79.
  • 56. P. Lévy, Sur certains processus stochastiques homogènes, Compositio Math. 7 (1939), 283-339. MR 1:150a
  • 57. -, Wiener's random function and other Laplacian random functions, Second Symposium of Berkeley. Probability and Statistics, U.C. Press, 1951, pp. 171-186. MR 13:476b
  • 58. -, Processus stochastiques et mouvement brownien, Gauthier-Villars, Paris, 1965, (first ed. 1948). MR 32:8363; MR 10:551a
  • 59. X.-J. Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory 65 (1997), no. 2, 325-333. MR 98d:11101
  • 60. P. Michel, Répartition des zéros de fonctions $L$ et matrices aléatoires, Séminaire Bourbaki, Exposé 887, 2001.
  • 61. Ph. Nanopoulos, Loi de Dirichlet sur ${N}\sp{\ast} $ et pseudo-probabilités, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 22, Aiii, A1543-A1546. MR 51:10272
  • 62. C.M. Newman, Fourier transforms with only real zeros, Proc. Amer. Math. Soc. 61 (1976), 245-251. MR 55:7944
  • 63. -, The GHS inequality and the Riemann hypothesis, Constr. Approx. 7 (1991), 389-399. MR 92f:11120
  • 64. A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273-308. MR 88d:11082
  • 65. -, The $10^{22}$-nd zero of the Riemann zeta function, available at www.research.att.com/$\sim$amo/, 2000.
  • 66. S. J. Patterson, An introduction to the theory of the Riemann Zeta-Function, Cambridge Univ. Press, Cambridge, 1988. MR 89d:11072
  • 67. J. Pitman, Cyclically stationary Brownian local time processes, Probab. Th. Rel. Fields 106 (1996), 299-329. MR 98d:60152
  • 68. -, The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest, Ann. Probab. 27 (1999), 261-283. MR 2000b:60200
  • 69. J. Pitman and M. Yor, A decomposition of Bessel bridges, Z. Wahrsch. Verw. Gebiete 59 (1982), 425-457. MR 84a:60091
  • 70. -, Dilatations d'espace-temps, réarrangements des trajectoires browniennes, et quelques extensions d'une identité de Knight, C.R. Acad. Sci. Paris t. 316, Série I (1993), 723-726. MR 93k:60208
  • 71. -, Decomposition at the maximum for excursions and bridges of one-dimensional diffusions, Itô's Stochastic Calculus and Probability Theory (N. Ikeda, S. Watanabe, M. Fukushima, and H. Kunita, eds.), Springer-Verlag, 1996, pp. 293-310. MR 98f:60153
  • 72. -, Random Brownian scaling identities and splicing of Bessel processes, Ann. Probab. 26 (1998), 1683-1702. MR 2000m:60097
  • 73. -, Laplace transforms related to excursions of a one-dimensional diffusion, Bernoulli 5 (1999), 249-255. MR 2000b:60190
  • 74. -, The law of the maximum of a Bessel bridge, Electronic J. Probability 4 (1999), Paper 15, 1-35. MR 2000j:60101
  • 75. -, Infinitely divisible laws associated with hyperbolic functions, Tech. Report 581, Dept. Statistics, U.C. Berkeley, 2000.
  • 76. G. Pólya, Verschiedene Bemerkungen zur Zahlentheorie, Jahber. Deutsch. Math. Vereinigung (1919), 31-40; Reprinted in Collected Papers, Vol III, MIT Press, Cambridge, Mass. 1984, pp. 76-85. MR 85m:01108a
  • 77. -, Elementarer Beweis einer Thetaformel, Sitz. Berich. Akad. Wissen. Phys.-math. Kl. (1927), 158-161; Reprinted in Collected Papers, Vol I, MIT Press, Cambridge, Mass. 1974, pp. 303-306. MR 58:21341
  • 78. -, Collected papers. Vol. II: Location of zeros (R. P. Boas, ed.), Mathematicians of Our Time, vol. 8, The MIT Press, Cambridge, Mass.-London, 1974. MR 58:21342
  • 79. A. Rényi and G. Szekeres, On the height of trees, J. Austral. Math. Soc. 7 (1967), 497-507. MR 36:2522
  • 80. D. Revuz and M. Yor, Continuous martingales and Brownian motion, Springer, Berlin-Heidelberg, 1999, 3rd edition. MR 2000h:60050
  • 81. B. Riemann, Über die Anzahl der Primzahlen unter eine gegebener Grösse, Monatsber. Akad. Berlin (1859), 671-680, English translation in [28].
  • 82. L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Vol. 1: Foundations, Wiley, 1994, 2nd. edition. MR 96h:60116
  • 83. J.-P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973, Translated from the French, Graduate Texts in Mathematics, No. 7. MR 49:8956
  • 84. Z. Shi and M. Yor, On an identity in law for the variance of the Brownian bridge, Bull. London Math. Soc. 29 (1997), no. 1, 103-108. MR 97k:60224
  • 85. G. R. Shorack and J. A. Wellner, Empirical processes with applications to statistics, John Wiley & Sons, New York, 1986. MR 88e:60002
  • 86. N. V. Smirnov, On the estimation of the discrepancy between empirical curves of distribution for two independent samples, Bul. Math. de l'Univ. de Moscou 2 (1939), 3-14, (in Russian).
  • 87. L. Smith and P. Diaconis, Honest Bernoulli excursions, J. Appl. Probab. 25 (1988), 464 - 477. MR 89m:60163
  • 88. J. Sondow, Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series, Proc. Amer. Math. Soc. 120 (1994), no. 2, 421-424. MR 94d:11066
  • 89. R. Stanley, Enumerative combinatorics, vol. 2, Cambridge University Press, 1999. MR 2000k:05026
  • 90. L. Takács, Remarks on random walk problems, Publ. Math. Inst. Hung. Acad. Sci. 2 (1957), 175-182. MR 21:929
  • 91. E.C. Titchmarsh, The Theory of the Riemann Zeta-Function. 2nd edition, revised by D.R. Heath-Brown, Clarendon Press, Oxford, 1986. MR 88c:11049
  • 92. P. Vallois, Amplitude du mouvement brownien et juxtaposition des excursions positives et négatives, Séminaire de Probabilités XXVI, Springer-Verlag, 1992, Lecture Notes in Math. 1526, pp. 361-373. MR 94j:60152
  • 93. -, Decomposing the Brownian path via the range process, Stoch. Proc. Appl. 55 (1995), 211-226. MR 96a:60067
  • 94. 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 87e:11102
  • 95. K. van Harn and F. W. Steutel, Infinite divisibility and the waiting-time paradox, Comm. Statist. Stochastic Models 11 (1995), no. 3, 527-540. MR 96g:60025
  • 96. W. Vervaat, A relation between Brownian bridge and Brownian excursion, Ann. Probab. 7 (1979), 143-149. MR 80b:60107
  • 97. G. S. Watson, Goodness-of-fit tests on a circle, Biometrika 48 (1961), 109-114. MR 24:A1777
  • 98. D. Williams, Decomposing the Brownian path, Bull. Amer. Math. Soc. 76 (1970), 871-873. MR 41:2777
  • 99. -, Path decomposition and continuity of local time for one dimensional diffusions. I, Proc. London Math. Soc. (3) 28 (1974), 738-768. MR 50:3373
  • 100. -, Diffusions, markov processes, and martingales, vol. 1: Foundations, Wiley, Chichester, New York, 1979. MR 80i:60100
  • 101. -, Brownian motion and the Riemann zeta-function, Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh, eds.), Clarendon Press, Oxford, 1990, pp. 361-372. MR 91h:60094
  • 102. M. Yor, Some aspects of Brownian motion, Part I: Some special functionals, Lectures in Math., ETH Zürich, Birkhäuser, 1992. MR 93i:60155
  • 103. -, Some aspects of Brownian motion, Part II: Some recent martingale problems, Lectures in Math., ETH Zürich, Birkhäuser, 1997. MR 98e:601407

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Additional Information

Philippe Biane
Affiliation: CNRS, DMA, 45 rue d’Ulm 75005 Paris, France
Email: Philippe.Biane@ens.fr

Jim Pitman
Affiliation: Department of Statistics, University of California, 367 Evans Hall #3860, Berkeley, CA 94720-3860
Email: pitman@stat.Berkeley.EDU

Marc Yor
Affiliation: Laboratoire de Probabilités, Université Pierre et Marie Curie, 4 Place Jussieu F-75252, Paris Cedex 05, France

DOI: https://doi.org/10.1090/S0273-0979-01-00912-0
Keywords: Infinitely divisible laws, sums of independent exponential variables, Bessel process, functional equation
Received by editor(s): October 1, 1999
Received by editor(s) in revised form: January 29, 2001
Published electronically: June 12, 2001
Additional Notes: Supported in part by NSF grants DMS-97-03961 and DMS-00071448
Article copyright: © Copyright 2001 American Mathematical Society

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