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This paper reviews known results which connect Riemann's in- tegral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of in- dependent 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.

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