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(可能包含多个变量)，它有多少个整数解？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，这个问题被证明是极其困难的。例如，即使当方程是两个变量的三次多项式方程时，我们也不知道如何回答。

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.