Multiple Dirichlet Series and Number Theory

多重狄利克雷级数和数论

• 批准号: 1601289
• 负责人: Gautam Chinta
• 金额: $ 16.8万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601289&HistoricalAwards=false
• 关键词: Multiple; Dirichlet; Series; Number; Theory

This project concerns research in number theory. Classical problems in number theory include questions about the distribution of prime numbers, representations of a number as a sum of squares, and counting the number of lattice points in a circle or polygon in the plane. These questions have applications to such varied fields as cryptography, integer programming, complexity theory, and numerical integration. This research project will study several related questions through the unifying perspective of Dirichlet series in several complex variables. Integrated with the research activities will be continued mentorship of high school, undergraduate, and graduate students. The investigator will introduce these students to research-level mathematics through supervised research projects.This research project will investigate number-theoretic problems in four different areas, unified by the common theme of naturally occurring multiple Dirichlet series. The research in the first area, zeta functions of prehomogeneous vector spaces (PVS), is particularly timely due to the recent resurgence of interest in this field. A second topic is the study certain period integrals of Eisenstein series. Arithmetic consequences include explicit formulas for representation numbers of quadratic forms, which can in turn be reinterpreted as formulas for counting integral points on certain flag varieties. In a third direction, hyperbolic Fourier expansions of Eisenstein series motivate the construction of different type of multiple Dirichlet series which count totally positive elements in the ring of integers of a totally real number field. This can be reinterpreted as a lattice point counting problem in irrational polytopes. Fourthly, the project will study new types of subgroup growth and subring growth zeta functions. While this work should stimulate new research in subgroup growth, the initial motivation for study of these objects comes from complexity theory and lattice cryptography.

该项目涉及数论研究。数论中的经典问题包括有关素数分布、将数字表示为平方和以及计算平面中圆形或多边形中格点数量的问题。 这些问题适用于密码学、整数规划、复杂性理论和数值积分等不同领域。 本研究项目将通过多个复杂变量的狄利克雷级数的统一视角来研究几个相关问题。 与研究活动相结合的是对高中生、本科生和研究生的持续指导。研究者将通过监督研究项目向这些学生介绍研究级数学。该研究项目将研究四个不同领域的数论问题，这些问题由自然发生的多个狄利克雷级数的共同主题统一起来。第一个领域，即预齐次向量空间 (PVS) 的 zeta 函数，由于最近人们对该领域的兴趣重新兴起，因此显得尤为及时。 第二个课题是研究爱森斯坦级数的某些周期积分。算术结果包括二次形式表示数的显式公式，这些公式又可以被重新解释为计算某些标志变量上的积分点的公式。在第三个方向上，爱森斯坦级数的双曲傅里叶展开激发了不同类型的多重狄利克雷级数的构造，该级数在全实数域的整数环中计算全正元素。这可以被重新解释为无理多胞体中的格点计数问题。第四，该项目将研究新型子群增长和子环增长zeta函数。 虽然这项工作应该会刺激子群增长的新研究，但研究这些对象的最初动机来自复杂性理论和格密码学。