Zeta-Functions, L-Functions, and Random Matrix Theory

研究者和他的同事正在研究分析数论中的问题，主要集中在Riemann-zeta函数和l -函数，以及它们的零和素数之间的关系。两个项目的重点是最近发现的随机矩阵理论在ζ函数中的显著应用。基廷和斯奈思的ζ函数特征多项式模型为解析数论中许多以前难以解决的问题提供了启发式的答案。然而，该模型的一个严重缺点是它不包含素数。相反，它们必须以特别的方式插入到每个新应用程序中。研究者和他的同事们有一个新的模型，以一种最自然的方式明确地整合了质数和零。他们把它应用到各种各样的问题中，比如对函数的矩和阶的估计，他们希望它能给他们提供关于零和素数之间联系的新见解。一个相关的项目探讨了零上的高斯酉系综猜想，素数和几乎素数的分布，以及涉及黎曼ζ函数的中值定理之间的关系。他们还研究了ζ函数的几何方面，比如它在临界线附近的曲率，以及零点间隙的大小。最后两个问题是数论中完全不同的领域。这些问题涉及有限域的乘群中元素的加性模式以及包含乘性特征的不完全指数和的值分布问题。这是数学领域的一个项目，叫做数论。数论的基本问题与数字的结构有关，特别是质数，因为它们是算术的基本组成部分，因此也是许多数学的基本组成部分。这一领域的许多最重要的问题都是如此棘手，甚至不可能猜出正确答案。最近，理论物理学家和分析数论家之间建立了一种非凡的合作关系，成功地回答了其中的一些问题。这次合作的核心是一个叫做黎曼ζ函数的模型，这是一个特殊的数学函数，一个多世纪以来，人们都知道它的性质包含了大量关于质数的信息。该模型基于随机矩阵理论，随机矩阵是以前用于模拟复杂物理系统（如重核）的对象。虽然这个模型相当成功，但它有一个严重的缺点，即不包含质数，而质数毕竟是主要的研究对象。研究者和他的同事们现在已经开发出一个模型，以最自然的方式整合质数和零，这个项目的主要目标是探索我们的新模型的进一步应用。一个相关的项目涉及一个被广泛相信的关于ζ函数零点的猜想的算术和分析结果。利用该领域的最新发展，我们还研究了ζ函数的几何方面，如它的曲率和零点之间的间隙大小。数论的两个不同领域的问题涉及有限域的结构，它们在编码理论和密码学中具有重要的应用。

DMS-0201457 Steve GonekAbstractThe investigator and his colleagues are studying problems in analyticnumber theory centered mainly on the Riemann-zeta function and L-functions,and relations between their zeros and the primes. Two projects focus on theremarkable, recently discovered applications of random matrix theory tothe zeta-function. Keating and Snaith's characteristic polynomial model ofthe zeta-function is providing heuristic answers to many previouslyintractable problems in analytic number theory. However, a serious drawbackof the model is that it does not contain the primes. Instead, they have tobe inserted in an ad hoc manner with each new application. The investigatorand his colleagues have a new model that explicitly integrates the primesand zeros in a most natural way. They are applying it to a variety ofproblems, such as moment and order estimates for the zeta-function, andthey expect it to give them new insights into the connections between thezeros and the primes. A related project explores the relations between theGaussian Unitary Ensemble Conjecture on the zeros, the distribution ofprimes and almost-primes, and mean-value theorems involving the Riemannzeta-function. They also study geometrical aspects of the zeta-function,such as its curvature near the critical line, and the size of gaps betweenzeros. Two final problems lie in an altogether different area of numbertheory. These concern additive patterns of elements in the multiplicativegroup of a finite field and the related question of the value distributionof incomplete exponential sums containing multiplicative characters.This is a project in the area of mathematics known as number theory. The fundamental questions of interest in number theory have to do with the structure of numbers, and in particular the prime numbers, as these are the fundamental building blocks of arithmetic and, therefore, of much of mathematics. Many of the most important questions in this area are so intractable that it is impossible even to guess correct answers to them. Recently, a remarkable partnership has developed between theoretical physicists and analytic number theorists, which is succeeding in answering some of these questions. At the center of this collaboration is a model of something called the Riemann zeta-function, which is a special mathematical function known for over a century to encode within its properties a great deal of information about prime numbers. This model is based on the theory of random matrices, objects previously used to model complicated physical systems such as heavy nuclei. Although the model has been quite successful, it has the serious drawback of not containing the prime numbers which, after all, are the principal objects of interest. The investigator and his colleagues have now developed a model that does integrate the primes and zeros in a most natural way, and a main goal of this project is to explore further applications of our new model. A related project concerns arithmetic and analytic consequences of a widely believed conjecture about the zeros of the zeta-function. Using recent developments in the field, we also investigate geometrical aspects of the zeta-function such as its curvature, and gap sizes between its zeros. Two problems in a different area of number theory concern the structure of finite fields, objects with important applications to coding theory and cryptography.