Problems in Analytic Number Theory

解析数论中的问题

• 批准号: 0070720
• 负责人: Hugh Montgomery
• 金额: $ 15万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070720&HistoricalAwards=false
• 关键词: Problems; Analytic; Number; Theory

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.

素数理论中最持久的问题之一是确定当一组数被筛选时还剩下多少元素。 有一个largeliterature关于这个话题，起源于开创性的论文Viggo Brun在1914年，但即使在今天，最知名的界限要么不是最优的或不被证明是最优的。 通过从一个简单的筛选情况开始，然后逐步移动到更复杂的配置，希望可以找到最优的边界，伴随着最优性的证明。 过去的工作对PI的Riemann zeta函数的零点的配对相关已被解释为提供证据，零点是spectralin性质。 配对相关猜想本身等价于一个关于素数在短时间间隔内均方分布的断言。Soundararajan，它被建议扩展到其他时刻的二阶矩，从而发展有关的分布函数的素数在短时间间隔的算法。 人们希望，这些新的信息，当解释zeta函数的零点时，将提供关于零点分布的进一步见解，包括黎曼假设。素数的看似不规则的分布已经困扰了数学家许多世纪。 在世纪早期，引入了newideas，它允许人们将筛选素数作为线性规划问题来处理。 这导致了许多新的结果，但直到今天，线性规划极值仍然在大多数情况下被发现。 通过从一个简单的情况开始，逐步地向更复杂的情况发展，人们希望最终能够找到极端的构型。 启发式concerningdistribution的素数在短时间内可以从thHardy-Littlewood素数k元组猜想，并从这样的推理获得的见解有影响的其他方面的素数理论，包括著名的黎曼假设，日期从1860年。