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

This paper reviews known results which connect Riemann's integral represent ations of his zeta function, involving Jacobi's theta function and its deri vatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian m otion and to higher dimensional Bessel processes. We present some character izations of these probability laws, and some approximations of Riemann's ze ta function which are related to these laws.