Cramér model

The Cramér model selects integers $n\ge 3$ independently with probability $1/\log n$, and conventionally sets $U_1=0$, $U_2=1$. It is a random model for the distribution of primes, not an assertion of independence for actual prime numbers.

If $P_n$ are the increasing selected integers in the Cramér model, then almost surely ${\mathrm{limsup}}_n(P_{n+1}-P_n)/(\log P_n{)}^2\le 1$. A zero block of length $(1+ϵ)(\log k{)}^2$ after $k$ has summable probability, so the Borel-Cantelli first lemma excludes these blocks eventually.