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

Biane, Philippe; Pitman, Jim; Yor, Marc

doi:10.1090/S0273-0979-01-00912-0

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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
Posted: 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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