Extreme Value Statistics in Probabilistic Number Theory

极值统计学是研究自然和社会系统中发生的罕见事件的科学。这种罕见事件的出现可以通过研究组成系统的组分之间相互作用产生的特定机制来理解。事实证明，数论中的一大类函数是扩展当前极值统计理论的强大模型。这个项目的目的是开发新的概率技术来描述在各种尺度上的数论函数极值的精细统计。这些技术将应用于物理和数学中的相关问题。该项目将为本科生和研究生提供计算机编程、数据科学和数学方面的研究培训机会。更准确地说，黎曼zeta函数（以及更大程度上的狄利克雷L函数）是数论中的一个基本函数，因为它编码整数中素数的分布。特别是这些函数的大值包含关于素数位置的信息，但它们仍然知之甚少。该研究项目建立在最近的进展，以描述在短时间间隔的zeta函数的大值。目前的项目旨在开发新的概率技术，以描述随着观测间隔的大小变化zeta函数的大值的精确统计。 特别是，在这个项目中进行的研究预计将揭示新的混合统计在小规模。该项目还寻求将技术扩展到Dirichlet L-函数，目前几乎没有结果。最终目标是建立更好的极值理论工具。该项目的跨学科性质预计将产生新的理论和数值工具的概率以及解析数论。理论结果和数值实验相结合，将澄清随机矩阵和L-functions.This奖项之间难以捉摸的连接反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Extreme value statistics is the study of rare events that occur in natural and social systems. The appearance of such rare events may be understood by investigating specific mechanisms arising from the interaction between the components constituting the system. It turns out that a large class of functions in number theory are formidable models to extend the current theory of extreme value statistics. The aim of this project is to develop new probabilistic techniques to describe at various scales the fine statistics of extrema of number-theoretic functions. These techniques will then be applied to related problems in physics and mathematics. The project will provide research training opportunities for undergraduate students and graduate students in computer programming, data science and mathematics.More precisely, the Riemann zeta function (and to a larger extent the Dirichlet L-functions) is a fundamental function in number theory as it encodes the distribution of prime numbers among the integers. Large values of such functions in particular contain information about the location of the primes, yet they remain poorly understood. This research project builds on recent progress to describe the large values of the zeta function in short intervals. The current project aims at developing new probabilistic techniques to describe the precise statistics of the large values of the zeta function as the size of the observation interval is varied. In particular, the research carried out in this project is expected to reveal new hybrid statistics at small scales. The project also seeks to extend the techniques to Dirichlet L-functions, where few results are currently available. The ultimate goal is to build better tools of extreme value theory. The interdisciplinary nature of the project is expected to produce new theoretical and numerical tools in probability as well as in analytic number theory. The theoretical results and numerical experiments in combination will clarify the elusive connection between random matrices and L-functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.