Arithmetic Statistics and Analytic 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（可能有许多变量），它有多少整数解？这个问题的一般答案将对数学，密码学和计算机科学领域产生巨大影响。然而，这个问题变得非常困难。例如，我们不知道如何回答它，即使方程是一个多项式方程的次数-3在两个变量。 如果我们考虑整个方程族，而不是只考虑一个方程，并询问解的平均数，那么这个问题就变得更容易处理了。研究一系列方程的解称为算术统计。 也可以从不同的方法来研究方程。也就是说，一些方程的解与复值函数的分析性质密切相关，称为zeta函数或L-函数。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。 在这两个领域的工作算术统计和解析数论，我的建议是联合收割机的工具，这两个主题，以解决基本问题，在这两个领域。这将对数论产生广泛的影响。

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.