Higher Correlations of Divisor Sums Related to Primes Iii: Small Gaps between Primes

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 4 + η times the average spacing between primes. Authors’ note. This paper was written in 2004, prior to the solution, in [8], of the problem considered here. In [8] it is shown that Δ = 0. While the main result in Theorem 1 has now been superseded, we believe the method used here is both of interest and future utility in other applications. In particular, the work of Green and Tao [12] on arithmetic progressions of primes makes use of Proposition 1 of this paper.