Theory of L-functions, prime numbers and divisors

本课题研究了l函数、素数和整数除数的零点分布。素数的分布和l函数的零的分布之间的关系早已为人所知，但有些关系直到最近才被发现。与A. Zaharescu和K. Soundararajan一起，研究者正在探索l函数0的虚部的分数部分之间的联系，短间隔内素数的计数，以及l函数0的对相关。第二个研究领域涉及揭示等差数列中具有精确k质因数（k1）的数字分布中的微妙不平等，这些问题与狄利克雷l函数的零点位置密切相关。研究的一个重点是检验质数（k=1）和“几乎质数”（k1）的等差数列行为的差异。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。研究作为除数基础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——它们如何被写成质数和一般的乘积。关于具有某种乘法结构的手指分布的问题，特别是素数，通常是非常困难和难以理解的。本项目正在研究这类问题中的几个问题，特别强调与称为l函数的特殊函数的行为以及与概率论和统计理论的联系。

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.