Problems in Analytic Number Theory

素数理论中最持久的问题之一是确定当一组数字被筛选后还剩下多少元素。关于这个话题有大量的文献，起源于维戈·布朗在1914年的开创性论文，但即使在今天，最著名的边界要么不是最优的，要么没有被证明是最优的。通过从一个简单的筛分情况开始，然后逐渐移动到更复杂的配置，希望可以找到最优界，并伴随着最优性的证明。过去关于黎曼ζ函数零点对相关性的PI研究被解释为提供了零点具有光谱性质的证据。对相关猜想本身等价于一个关于短间隔内素数均方分布的断言。在与k. Soundararajan的新工作中，提出将二阶矩性推广到其他矩，从而发展了关于短间隔内质数分布函数的启发式。希望这些新的信息，当用ζ函数的零点解释时，将提供关于零点分布的进一步见解，包括黎曼假设。素数看似不规则的分布，几个世纪以来一直困扰着数学家。在20世纪早期，引入了一些新的思想，使人们能够处理一个线性规划的问题——筛分素数。这导致了许多新的结果，但即使在今天，在大多数情况下仍然可以找到线性规划的极值。通过从一个简单的情况开始，逐渐移动到更复杂的情况，希望最终有可能找到极端构型。关于短间隔内素数分布的启发式可以从hardy—Littlewood素数k元组猜想中发展出来，并且从这种推理中获得的见解对素数理论的其他方面产生了影响，包括1860年著名的黎曼假设。

One of the most enduring problems of prime number theory is to determinehow many elements remain when a set of numbers is sieved. There is a largeliterature on this topic, originating in seminal papers of Viggo Brun in 1914,but even today the best known bounds are either not optimal or not provedto be optimal. By starting with a simple sieving situation and then movingincrementally to more complicated configurations, it is hoped that optimalbounds can be found, accompanied by proofs of optimality. The past workon the PI on the pair correlation of the zeros of the Riemann zeta functionhas been interpreted as providing evidence that the zeros are spectralin nature. The Pair Correlation Conjecture itself is equivalent to anassertion concerning the mean square distribution of primes in short intervals.In new work with k. Soundararajan, it is proposed to extend the second momentheuristics to other moments, and hence develop heuristics concerning thedistribution function of primes in short intervals. It is hope that thisnew information, when interpreted in terms of zeros of the zeta function,will provide further insights concerning the distribution of the zeros,including the Riemann Hypothesis.The seemingly irregular distribution of prime numbers has been a puzzleto mathematicians for many centuries. In the early 20th century, newideas were introduced, which allowed one to deal with sieving for primesas a problem of linear programming. This led to many new results, buteven today the linear programming extremals remain to be found in mostsituations. By starting with a simple situation and moving incrementallyto more complicated ones, it is hoped that it will at last bepossible to locate the extremal configurations. Heuristics concerningthe distribution of primes in short intervals can be developed from theHardy--Littlewood prime k-tuple conjecture, and the insights gained fromsuch reasoning has an impact on other aspects of prime number theory,including the famous Riemann Hypothesis which dates from 1860.