Theory of L-functions, prime numbers and divisors

本课题主要研究L函数的零点分布、素数和整数的因子。素数和L函数的零点分布之间的关系早已为人所知，但一些关系是最近才被发现的。与A.Zaharesu和K.Soundararajan一起，研究了L函数的零点的虚部的小数部分、短区间素数的计数和L函数的零点对相关之间的联系。第二个研究领域涉及发现算术级数中具有精确k个素数因子(K1)的数的分布中的微妙不平等，这些问题与狄里克莱特L函数的零点的位置密切相关。研究的焦点是考察素数(k=1)和几乎素数(K1)在算术级数中的行为的差异。第三，研究人员将继续研究“典型”和“非典型”整数的除数是如何分布的。对研究除数的概率理论的研究将是一个主要主题。正整数可能是数学中最基本的对象，理解它们的乘法结构很重要--它们如何写成素数和一般的乘积。关于具有一定乘法结构的整数的分布问题，特别是元数的分布问题，往往是一个非常困难和难以理解的问题。在这个项目中，我们正在研究几个这类问题，特别是与称为L函数的特殊函数的性质的联系，以及与概率统计理论的联系。

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.