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Proceedings of the London Mathematical Society Advance Access originally published online on July 16, 2007
Proceedings of the London Mathematical Society 2007 95(3):653-686; doi:10.1112/plms/pdm021
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© 2007 London Mathematical Society

Higher correlations of divisor sums related to primes III: small gaps between primes

D. A. Goldston

Department of Mathematics
San Jose State University
San Jose, CA 95192
USA

C. Y. Yildirim

Department of Mathematics
Bogaziçi University
Bebek
Istanbul, 34342
Turkey
and
Feza Gürsey Enstitüsü
Çengelköy
Istanbul, P.K. 6, 81220
Turkey
yalciny{at}boun.edu.tr

Received 8 November 2005. Revision received 17 March 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 {eta} > 0, a positive proportion of consecutive primes are within Formula times the average spacing between primes.

2000 Mathematics Subject Classification 11N05 (primary), 11P32 (secondary).

The research of Goldston was supported by NSF; that of Yildirim was supported by TÜBÏTAK


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