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.

素数是整数的乘法积木，了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Riemann Zeta函数，这是解析数论领域的基础追求。特别是，彻底了解该函数的所谓非平凡零点的位置将给出直到给定(大)整数的素数的非常精确的渐近公式。这是所有数学中的一个中心问题，与其他深层问题有联系，比如最初由高斯研究的班数问题。这项工作旨在进一步探索Riemann Zeta-函数和更一般的L-函数的分析性质，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，拟议的工作还包括为解析数论的活跃研究领域提供新颖、全面的指导的“路径项目”，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理论，分为三个主题。第一个主题涉及Riemann Zeta函数的零点的垂直分布。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方法的限制。此外，还将编写一本关于Riemann Zeta函数的零点之间的间隙问题的综合指南，以及它与解析数论中几个活跃的研究领域的联系。在第二个主题中，我们将探讨切博塔雷夫密度定理的应用。特别地，我们将证明某些规定族中“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机制和应用。特别地，渐近大筛将被用来研究各种L函数的零点的分布。这项技术将得到加强，并随后应用于某些L函数的矩的计算方面取得新的进展。最后，将编写渐近大筛子的全面指南，这将在各种应用中发挥作用，使该技术更广为人知和理解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。