One can use the Riemann zeta function evaluated at a parameter s>1 to create a probability distribution on the positive integers. If X(s) is a random integer with this distribution, one might ask whether one can produce a natural stochastic process in the parameter s. Using an idea of Lloyd this is possible and reveals a predominant Poisson behavior in X(s). In addition, we can use mod-Poisson convergence of Jacod, Kowalski and Nikeghbali to prove precise large deviation estimates for the number of prime divisors of X(s) as s goes down to 1. These ideas apply more generally to integers selected via Dirichlet series, polynomials with coefficients in a finite field or ideals selected in a Dedekind domain. This talk is based on joint work with Jingyuan Chen and Mariia Khodiakova.