CAREER: Research in and Pathways to Analytic Number Theory

素数是整数的乘法构建块，理解它们的性质是数论的中心主题。理解它们在整数中的分布的一种方法是通过研究黎曼zeta函数，这是解析数论领域的基础。特别是，彻底理解这个函数的所谓非平凡零点的位置将给出非常精确的素数数量的渐近公式，直到一个给定的（大）整数。这是一个中心问题，在所有的数学与连接到其他深层次的问题，如类数问题，最初研究的高斯。拟议的工作旨在进一步探索黎曼zeta函数的分析性质，更一般地说，L函数，其总体目标是获得有关这些函数的零点的新信息。除了研究目标外，拟议的工作还包括“路径项目”，为解析数论的积极研究领域提供新颖，全面的指导，以及旨在增加历史上代表性不足的群体在STEM中的参与的本科教育计划。 这个项目的研究目标是解析数论，分为三个主题。第一个主题是关于黎曼zeta函数零点的垂直分布。特别是，将确定用于检测非平凡零之间的间隙波动的最先进的方法的限制。此外，一个全面的指导问题之间的差距的黎曼zeta函数和它的连接到几个活跃的研究领域，在解析数论将被写入。在第二个主题中，将探讨Chebotarev密度定理的应用。特别是，改进的零密度估计的“大多数”L-函数在某些规定的家庭将被证明。第三个主题包括渐近大筛的力学和应用。特别地，渐近大筛将被用来研究各种L-函数的零点的分布。加强技术将开发和随后应用于取得新的进展，计算某些L-函数的时刻。最后，一个全面的指南，渐近大筛，这是准备在各种应用中有用，将被写入，使技术更广泛地了解和理解。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Primes are the multiplicative building blocks of integers, and understanding their properties is a central theme in number theory. One way to understand their distribution among the integers is through the study of the Riemann zeta-function, a pursuit that is foundational to the area of analytic number theory. In particular, a thorough understanding of the location of the so-called nontrivial zeros of this function would give very precise asymptotic formulas for the number of primes up to a given (large) integer. This is a central problem in all of mathematics with connections to other deep problems, such as the class number problem, originally studied by Gauss. The proposed work seeks to further explore the analytic properties of the Riemann zeta-function and, more generally, of L-functions, with an overarching goal to obtain new information regarding the zeros of these functions. In addition to the research objectives, the proposed work includes "Pathway Projects" to provide novel, comprehensive guides to areas of active research in analytic number theory, and an undergraduate educational program aimed at increasing the participation of historically underrepresented groups in STEM. The research objectives of this project are in analytic number theory and fall into three themes. The first theme concerns the vertical distribution of zeros of the Riemann zeta-function. In particular, limitations on the state-of-the-art methods used to detect fluctuations in gaps between non-trivial zeros will be determined. Moreover, a comprehensive guide to the problem of gaps between zeros of the Riemann zeta-function and its connection to several active areas of research in analytic number theory will be written. In the second theme, applications of the Chebotarev density theorem will be pursued. In particular, improved zero-density estimates for "most" L-functions within certain prescribed families will be proved. The third theme encompasses the mechanics and applications of the asymptotic large sieve. In particular, the asymptotic large sieve will be used to study the distribution of the zeros of various L-functions. A strengthening of the technique will be developed and subsequently applied to make new progress on calculating moments of certain L-functions. Finally, a comprehensive guide to the asymptotic large sieve, which is poised to be useful in various applications, will be written to make the technique more widely known and understood.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.