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Mathematical Sciences: Multiplicative and Additive Theory ofPrime Numbers and Related Topics in Analytic Number Theory

数学科学：素数的乘法和加法理论以及解析数论中的相关主题

基本信息

批准号：
8705710
负责人：
Daniel Goldston
金额：
$ 3.17万
依托单位：
San Jose State University Foundation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1987
资助国家：
美国
起止时间：
1987-06-01 至 1989-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=8705710&HistoricalAwards=false
关键词：
Mathematical Sciences Multiplicative Additive Theory

项目摘要

This research concerns prime number theory, which can be divided into two approaches, the multiplicative and the additive. The multiplicative theory is largely concerned with the duality between primes and zeros of the Riemann zeta function. The additive theory is based on exponential sums of primes and typically uses characters and Dirichlet L-functions in order to express additive characters as sums of multiplicative functions. The overlap between these two areas is what is being studied in this research. Some specific topics to be examined are pair correlation of zeros of zeta functions, prime pairs and their connection with Kloostermann sums and Barban-Davenpoort type theorems, small gaps between zeta zeros, the function S(T) and Siegal zeros. This research concerns prime number theory, that is, the distribution of the prime numbers in the integers. Certain complex analytic functions, called zeta functions, contain this information and so they are studied uses tools from analysis. Thus this subject is a nice blending of number theory and analysis. The results are very deep and difficult to come by. Goldston has been remarkably successful in solving problems in this area. This research will certainly push forward our knowledge substantially in this important area on mathematics.

本研究涉及素数论，可分为乘法和加法两种方法。乘法理论主要关注黎曼 zeta 函数的素数和零点之间的对偶性。加法理论基于素数的指数和，并且通常使用特征和狄利克雷 L 函数，以便将加法特征表示为乘法函数的和。本研究正在研究这两个领域之间的重叠。要检查的一些具体主题包括 zeta 函数零点的配对相关性、素数对及其与 Kloostermann 和和 Barban-Davenpoort 型定理的联系、zeta 零点之间的小间隙、函数 S(T) 和 Siegal 零点。这项研究涉及素数论，即素数在整数中的分布。某些复杂的分析函数（称为 zeta 函数）包含此信息，因此可以使用分析工具来研究它们。因此，这门学科很好地融合了数论和分析。结果非常深刻，也很难得到。戈德斯顿在解决这一领域的问题方面取得了巨大成功。这项研究无疑将大大推进我们在数学这一重要领域的知识。