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

Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions
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by Philippe Biane, Jim Pitman and Marc Yor PDF

Bull. Amer. Math. Soc. 38 (2001), 435-465 Request permission

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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