Multiple Dirichlet Series and Number Theory

这个项目涉及数论的研究。数论中的经典问题包括质数的分布，数的平方和表示，以及平面上圆或多边形中点阵的个数。这些问题在密码学、整数规划、复杂性理论和数值积分等不同领域都有应用。本研究计划将透过狄利克雷级数在多个复杂变数中的统一视角，研究若干相关问题。与研究活动相结合，将继续指导高中生，本科生和研究生。研究者将通过监督的研究项目向这些学生介绍研究水平的数学。本研究计划将探讨四个不同领域的数论问题，这些问题由自然发生的多重狄利克雷级数的共同主题统一。第一个领域的研究，预齐次向量空间（PVS）的zeta函数，由于最近对这一领域的兴趣复苏而特别及时。第二个课题是研究爱森斯坦级数的某些周期积分。算术结果包括二次型表示数的显式公式，这些公式反过来可以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立叶展开式激发了在全实数域的整数环上计算全正元素的不同类型的多重狄利克雷级数的构造。这可以重新解释为无理性多面体中的点阵计数问题。第四，研究新型子群增长和子群增长zeta函数。虽然这项工作应该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点阵密码学。

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.