RUI: Distribution of Primes and a Higher Correlation Method

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Many questions on primes can be addressed by the use of short divisor sum approximations. These sums arise in the circle method, but they can be used independently of the circle method. They may also be viewed as truncations of Ramanujan expansions. The main terms in many asymptotic formulas for primes result from summing these short divisor sums into a singular series. These singular series therefore also reflect the properties of primes, and it is often an interesting and non-trivial problem to evaluate formulas involving singular series. Starting in 1990, the principal investigator has been working on obtaining lower bounds for problems involving primes. Up to now all this work has been on the binary correlation of short divisor sum approximations for primes. The investigator and C. Yildirim intend to extend this earlier work to higher correlations, and apply the results to obtain unconditional lower bounds for higher moments of primes in short intervals. The method also has application to other problems in analytic number theory. In addition, some questions on power sums and other problems will be examined in collaboration with undergraduate and graduate students at San Jose State University.The distribution of prime numbers was first studied by the Greeks over two thousand years ago. Many significant results have been proved, but many difficult problems remain to be solved. With computers one can verify many conjectures about the distribution of primes with startling precision, and yet proofs of these conjectures are beyond our current state of knowledge. The principal investigator has focused his research on the problem of proving that the gap between consecutive primes will frequently be much smaller than the average gap size, and will infinitely often be smaller than any fraction of this average gap size. The techniques used to study this problem come from many areas of mathematics, statistics, and physics. Because of the fundamental role primes play in mathematics, it is to be expected that progress in this area will have applications in other areas.