CAREER: Research in and Pathways to Analytic Number Theory

素数是整数的乘法组成部分，理解它们的性质是数论的中心主题。了解它们在整数中的分布的一种方法是通过研究黎曼ζ函数，这是分析数论领域的基础。特别是，彻底理解这个函数的所谓非平凡零点的位置，将给出一个非常精确的直至给定（大）整数的素数的渐近公式。这是所有数学中的一个中心问题，它与其他深层次问题有联系，比如高斯最初研究的类数问题。提出的工作旨在进一步探索黎曼ζ函数的解析性质，更一般地说，是l函数的解析性质，其总体目标是获得关于这些函数的零点的新信息。除了研究目标之外，拟议的工作还包括“路径项目”，为分析数论的活跃研究领域提供新颖，全面的指导，以及旨在增加历史上代表性不足的群体参与STEM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ζ函数零点的垂直分布。特别是，将确定用于检测非平凡零点间隙波动的最先进方法的局限性。此外，黎曼ζ函数的零点间隙问题及其与分析数论中几个活跃研究领域的联系的综合指南将被编写。在第二个主题中，将探讨Chebotarev密度定理的应用。特别是，将证明在某些规定族内的“大多数”l函数的改进的零密度估计。第三个主题包括渐近大筛的力学和应用。特别地，渐近大筛将用于研究各种l -函数的零点分布。本文将进一步加强这一技术，并将其应用于计算某些l -函数的矩方面取得新的进展。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的，将编写使该技术更广泛地了解和理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.