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

DMS-0201457 Steve Gonek研究人员和他的同事们正在研究解析数论中的问题，主要集中在Riemann-Zeta函数和L函数，以及它们的零点和素数之间的关系。有两个项目专注于最近发现的随机矩阵理论在Zeta函数中的应用。基廷和斯奈斯的Zeta函数的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陷是它不包含素数。相反，它们必须以一种特别的方式插入到每个新的应用程序中。这位研究人员和他的同事们有了一个新的模型，它以最自然的方式显式地整合了素数和零点。他们正在将它应用于各种问题，例如Zeta函数的矩和阶数估计，他们希望它能让他们对零点和素数之间的联系有新的见解。一个相关的项目探索了关于零点的高斯酉系综猜想、素数和几乎素数的分布以及涉及Riemannzeta函数的中值定理之间的关系。他们还研究Zeta函数的几何方面，如临界线附近的曲率，以及零点之间的间隙大小。最后两个问题存在于数论的一个完全不同的领域。这些问题涉及有限域的乘法群中元素的加法模式以及含有乘法特征的不完全指数和的值分布的相关问题。这是数学领域中的一个被称为数论的项目。数论中感兴趣的基本问题与数的结构有关，尤其是质数，因为它们是算术的基本构件，因此也是许多数学的基础。这一领域的许多最重要的问题都非常棘手，甚至不可能猜出正确的答案。最近，理论物理学家和解析数学家之间发展了一种引人注目的合作伙伴关系，成功地回答了其中一些问题。这次合作的中心是一个名为Riemann Zeta-Function的模型，这是一种已知了一个多世纪的特殊数学函数，在其性质中编码了大量关于质数的信息。这个模型是基于随机矩阵的理论，随机矩阵是以前用来模拟重核等复杂物理系统的对象。虽然这个模型相当成功，但它有一个严重的缺陷，那就是它不包含素数，毕竟素数是主要的感兴趣的对象。这位研究人员和他的同事们现在已经开发出一种以最自然的方式整合素数和零点的模型，这个项目的一个主要目标是探索我们新模型的进一步应用。一个相关的项目涉及一个普遍认为的关于Zeta函数的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Zeta函数的几何方面，如它的曲率，以及它的零点之间的间隙大小。数论不同领域中的两个问题涉及有限域的结构，这两个问题在编码理论和密码学中有重要的应用。

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.