Primes in tuples II

Abstract

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.