UC Irvine

The Riemann Zeta distribution is one of many ways to sample a positive integer at random. Many properties of such a random integer X, like the number of distinct prime factors it has, are of concern. The zeta distribution facilitates the calculation of the probability of X having such and such property. For example, for any distinct primes p and q, the events {p divides X} and {q divides X} are independent. One cannot say this if instead, X were chosen randomly according to a geometric, poisson, or uniform distribution on the discrete interval [n] = {1,...n} for some n. Taking advantage of such facilities, we find a formula for the moment generating function of ω(X), and Ω(X), where ω(n) and Ω(n) are the usual prime counting functions in number theory. We use this to prove an Erdos-Kac like CLT for the number of distinct and total prime factors of such a random variable. Furthermore, we obtain Large Deviation results for these random variables, as well as a CLT for the number of prime factors in any arithmetic progression. We also investigate some divisibility properties of a Poisson random variable, as the rate parameter λ goes to infinity. We see that the limiting behavior of these divisibility properties is the same as in the case of a uniformly chosen positive integer from {1, .., n}, as n → ∞.