CAREER: Research in and Pathways to Analytic Number Theory

职业：解析数论的研究和途径

• 批准号: 2239681
• 负责人: Caroline Turnage-Butterbaugh
• 金额: $ 40万
• 依托单位: Carleton College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2239681&HistoricalAwards=false
• 关键词: CAREER; Research; Pathways; Analytic; Number

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.

素数是整数的乘法构建块，理解它们的属性是数字理论的中心主题。理解它们在整数之间的分布的一种方法是研究Riemann Zeta功能，这是分析数理论领域的基础的追求。特别是，对此功能的所谓非平凡零的位置有透彻的理解将为给定（大）整数的数量提供非常精确的渐近渐近公式。在所有数学中，这是一个与其他深层问题的联系，例如最初由高斯研究的班级问题问题。拟议的工作旨在进一步探索Riemann Zeta功能的分析特性，更普遍地是L功能的分析性能，其总体目标是获得有关这些功能零的新信息。除研究目标外，拟议的工作还包括“途径项目”，以为分析数理论的积极研究领域提供新颖的，全面的指南，以及旨在增加STEM中代表性不足群体的参与的本科教育计划。 该项目的研究目标是在分析数理论中，属于三个主题。第一个主题涉及Riemann Zeta功能的零的垂直分布。特别是，将确定用于检测非平凡零之间间隙波动的最新方法的局限性。此外，将编写一个关于Riemann Zeta功能的零零与与分析数理论中几个活跃研究领域的联系之间差距问题的综合指南。在第二个主题中，将追求Chebotarev密度定理的应用。特别是，将证明对某些规定家庭中“大多数”功能的零密度估计得到改善。第三个主题涵盖了渐近大筛子的力学和应用。特别是，渐近大筛将用于研究各种L功能的零的分布。该技术的加强将被开发，然后应用于在计算某些L功能的力矩上取得新的进展。最后，将编写一份有助于在各种应用中有用的渐近大筛子的综合指南，以使该技术更广泛地了解和理解。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准来通过评估来进行评估的支持。