Bulletin of the American Mathematical Society

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

Author(s): Philippe Biane; Jim Pitman; Marc Yor
Journal: Bull. Amer. Math. Soc. 38 (2001), 435-465.
MSC (2000): Primary 11M06, 60J65, 60E07
Posted: June 12, 2001
Retrieve article in: PDF

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.

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 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 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

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11M06, 60J65, 60E07

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: 10.1090/S0273-0979-01-00912-0
PII: S 0273-0979(01)00912-0
Keywords: Infinitely divisible laws, sums of independent exponential variables, Bessel process, functional equation
Received by editor(s): October 1999,
Received by editor(s) in revised form: January 29, 2001
Posted: June 12, 2001
Additional Notes: Supported in part by NSF grants DMS-97-03961 and DMS-00071448
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS