Arithmetic Statistics and Analytic Number Theory

算术统计与解析数论

• 批准号: RGPIN-2017-06589
• 负责人: Shankar, Arul
• 金额: $ 1.82万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=664863
• 关键词: Arithmetic; Statistics; Analytic; Number; Theory

***One of the most fundamental questions in Number Theory is: given an equation (in possibly many variables) with integer coefficients, how many integer solutions does it have? A general answer to this question will have a tremendous impact across the fields of Mathematics, Cryptography, and Computer Science. However, this question turns out to be extremely difficult. For example, we do not know how to answer it even when the equation is a polynomial equation of degree-3 in two variables.******The question becomes somewhat more manageable if instead of considering only one equation, we consider an entire family of equations, and ask about the average number of solutions. The study of solutions across a family of equations is called Arithmetic Statistics.******Equations may also be studied from a different approach. Namely, the solutions to some equations are intimately related to the analytical properties of complex valued functions, called zeta functions or L-functions. The study of such functions, and their relation to these equations is part of what is called Analytic Number Theory.******Having worked in both the fields of Arithmetic Statistics and Analytic Number Theory, my proposal is to combine the tools of these two subjects to tackle fundamental problems in both fields. This will have a broad impact across Number Theory.

***数论中最基本的问题之一是：给定一个具有整数系数的方程（可能有多个变量），它有多少个整数解？这个问题的一般性答案将对数学、密码学和计算机科学领域产生巨大影响。然而，事实证明这个问题非常困难。例如，即使方程是两个变量的 3 次多项式方程，我们也不知道如何回答。********如果我们不只考虑一个方程，而是考虑一个方程，那么这个问题会变得更容易管理。整个方程组，并询问解的平均数量。对一系列方程组的解的研究称为算术统计。******也可以通过不同的方法来研究方程。也就是说，某些方程的解与复值函数（称为 zeta 函数或 L 函数）的分析性质密切相关。对这些函数及其与这些方程的关系的研究是所谓的解析数论的一部分。******在算术统计和解析数论两个领域工作过之后，我的建议是将以下工具结合起来：这两个主题旨在解决两个领域的基本问题。这将对数论产生广泛的影响。