In this talk we will discuss properties of integers selected according to the Riemann zeta distribution. We will emphasize two aspects of this distribution. The first is its faithful similarity to properties of an integer chosen according to the uniform distribution on a finite interval. The second aspect will be the appearance of Poisson behavior under this distribution. The Riemann zeta function is given for $\mbox{Re} z>1$ by 

$$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}.$$

An alternative description is given by

$$\zeta(z)=\Pi_{p\in\mathcal{P}}\lt(1-\frac{1}{p^z}\rt)^{-1},$$

where $\mathcal{P}$ denotes the set of primes.

In our discussions we will replace the complex $z$ by a real number $s>1.$ We will denote by $X_s$ a random variable with the distribution

P(X_s=n)=\frac{1}{\zeta(s)n^s},\, n=1,2,3,\cdots.

The statistical properties of $X_s$ is the focus of the talk.

The talk is based on joint work with Adrien Peltzer.