Theory of L-functions, prime numbers and divisors

L 函数、素数和约数理论

• 批准号: 0555367
• 负责人: Kevin Ford
• 金额: $ 14.77万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555367&HistoricalAwards=false
• 关键词: Theory; functions; prime; numbers; divisors

This project is concerned with investigations into the distributionof zeros of L-functions, prime numbers and divisors of integers.The distribution of prime numbers and zeros of L-functions have longbeen known to be related, but some relations have only recently cometo light. Together with A. Zaharescu and K. Soundararajan, theinvestigator is exploring connections between fractionalparts of imaginary parts of zeros of L-functions, counts ofprime numbers in short intervals, and the pair correlation of zerosof L-functions. A second area of study involves uncoveringsubtle inequities in the distribution of numbers with exactlyk prime factors (k1) in arithmetic progressions, problems intimatelyconnected with the location of the zeros of Dirichlet L-functions.A focus of the research is examining differences between thethe behavior in arithmetic progressions of primes (k=1) and "almostprimes" (k1). Thirdly, the investigator will continue studyinghow the divisors of "typical" and "atypical" integers are distributed.Investigations into the probability theory underlying the study ofdivisors will be a major theme.The positive integers are perhaps the most basic objects inmathematics, and it is important to understand their multiplicativestructure - how they may be written as products, both ofprime numbers and in general. Questions about the distribution ofintegers with a certain multiplicative structure, especially primenumbers, are often very difficult and poorly understood.Several problems of this type are being investigated in this project, withparticular emphasis on the connections with the behavior ofspecial functions called L-functions and with probability andstatistical theory.

本课题研究L-函数的零点分布、素数分布和整数的除数分布，素数分布和L-函数的零点分布一直被认为是相关的，但有些关系是最近才被发现的。 与A. Zaharescu和K. Soundararajan，研究人员正在探索之间的联系fractional部分的虚部零的L-函数，计数素数在短时间内，和对相关的零的L-函数。 第二个研究领域是揭示算术级数中具有k个素数因子（k1）的数的分布中的微妙不等式，这些问题与Dirichlet L-函数的零点的位置密切相关。研究的重点是考察素数（k=1）和“几乎素数”（k1）算术级数的行为之间的差异。 第三，研究者将继续研究“典型”和“非典型”整数的除数是如何分布的。研究除数研究的概率论将是一个主要主题。正整数可能是数学中最基本的对象，理解它们的乘法结构是很重要的--它们如何被写成乘积，包括素数和一般的乘积。 关于具有某种乘法结构的整数的分布问题，特别是素数，通常是非常困难和难以理解的。这类问题中的几个问题在这个项目中正在研究，特别强调与称为L-函数的特殊函数的行为以及与概率和统计理论的联系。