RUI: Distribution of Primes and a Higher Correlation Method

许多关于素数的问题可以通过使用短因子和近似来解决。这些和出现在圆法中，但它们可以独立于圆法使用。它们也可以被看作是拉马努金展开式的截断。许多素数的渐近公式中的主要项是将这些短因子和求和为奇异级数的结果。因此，这些奇异级数也反映了素数的性质，并且评估涉及奇异级数的公式通常是一个有趣且非平凡的问题。 从1990年开始，首席研究员一直致力于获得涉及素数问题的下界。到目前为止，所有这些工作都是关于素数的短因子和近似的二元相关性。研究者和C. Yildirim打算将这一早期的工作扩展到更高的相关性，并将结果应用于获得短间隔内素数高阶矩的无条件下界。该方法也适用于解析数论中的其他问题。此外，还将与圣何塞州立大学的本科生和研究生合作研究幂和等问题。素数的分布最早是由希腊人在两千多年前研究的。许多有意义的结果已被证明，但许多难题仍有待解决。人们可以用计算机以惊人的精确度验证许多关于素数分布的假设，然而这些假设的证明超出了我们目前的知识水平。首席研究员将他的研究重点放在证明连续素数之间的差距经常会比平均差距大小小得多，并且会无限地经常小于这个平均差距大小的任何分数。用于研究这个问题的技术来自数学，统计学和物理学的许多领域。由于素数在数学中的基本作用，可以预期这一领域的进展将在其他领域得到应用。

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.