Higher correlations of divisor sums related to primes III: small gaps between primes - Goldston, D. A.; Y ld r m, C. Y. (Proceedings of the London Mathematical Society, 2007)
We use divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η > 0, a positive proportion of consecutive primes are within 14+η times the average spacing between primes.