Primes in tuples II

We prove that

$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $

where p _n denotes the nth prime. Since on average p _n+1?p _n is asymptotically log _n , this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p?p′ between primes which includes the small gap result above.