In the Princeton Companion of Mathematics, the Analytic Number Theory section, the author mentions what he calls Gauss-Kramer model, which is simply modeling the integers on a countable sequence of random variables $\{ X_{n} \}$, where the variable $X_{n}$ has probability of $1/log(n)$ of being 1 and otherwise 0. Each variable $X_{n}$ represents whether the number n is prime or not. The Gauss-Kramer model conjectures that statements about the distribution of primes correspond to the statements on the countable collection of random variables with probability 1 of being true.

Question: Is anyone aware of a good exposition of the Gauss-Kramer model and any attempt to make it more rigorous using Model Theory or through other routes?