RUI: Moments of Short Divisor Sums and the Distribution of Primes

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The investigator in joint work with C. Y. Yildirim has over the last three years determined the correlations of a short divisor sum which approximates the prime numbers. The corresponding correlations for primes were conjectured by Hardy and Littlewood in 1923 and are now referred to as the prime r-tuple conjecture. The proof of this conjecture is probably far beyond our current state of knowledge, but the corresponding result for the short divisor sum is now available. Further, using the Bombieri-Vinogradov theorem one can also obtain all the correlations when one of the divisor sums is replaced by the primes. These two types of correlations together with positivity allow us to obtain information about primes in the form of lower bounds on certain sums over primes. The first and main aim of this project is to apply these new correlation results to the problem of finding small gaps between primes. The main question to be answered is whether one can prove that there are infinitely many prime gaps shorter than any small multiple of the average gap size. At present it is only known that there are small gaps of size less than 0.248 times the average spacing. The method we use is based on moments for short divisor sums in short intervals. Preliminary work indicates we should be able to substantially improve on all previous results. There are many possible refinements and a variety of ways to apply the moment results which will be investigated. The second aim of this project is to refine the correlation results so far obtained and extend their range of applicability. A third aim is to seek further applications of the method to primes in arithmetic progressions and zeros of the Riemann zeta-function.This proposal is concerned with proving results on the distribution of primes. The primes have been a topic of interest since the Greeks who first proved both the infinitude of primes and the unique factorization of all integers into primes. The primes are intimately connected with the Riemann hypothesis and are fundamental objects in both number theory and mathematics. New results and methods concerning primes have been used in many areas of mathematics, physics, and computer science. To cite a recent example, the celebrated proof that primality testing can be done in polynomial time depended on the existence of primes p with p - 1 having a large prime factor, a result proved by Fouvry 17 years ago which up to this year had only an esoteric application to the first case of Fermat's last theorem.