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Bulletin of the American Mathematical Society

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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
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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Additional Information

Philippe Biane
Affiliation: CNRS, DMA, 45 rue d’Ulm 75005 Paris, France

Jim Pitman
Affiliation: Department of Statistics, University of California, 367 Evans Hall #3860, Berkeley, CA 94720-3860
MR Author ID: 140080
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

Keywords: Infinitely divisible laws, sums of independent exponential variables, Bessel process, functional equation
Received by editor(s): October 19, 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