Several aspects of L-functions

L-函数在数论中扮演着核心角色，因为该领域许多最深奥的问题都围绕着它们。它们编码算术对象的属性，例如质数。欧几里得在2300年前证明了素数有无穷多个，高斯在200年前猜想了素数的渐近公式。这个渐近公式最终由Hadamard和de la vallsame Poussin在1896年证明，并被称为素数定理。然而，获得一个像高斯猜想的那样精确的公式仍然是一个悬而未决的问题，因为它取决于黎曼假设（RH），这是克莱数学研究所获得100万美元奖金的七大千年问题之一。RH是关于黎曼函数的零点的表述，黎曼函数是最简单的l函数。当考虑更一般的l函数时，会出现许多方向。本主题中最基本的问题涉及到l函数可以有多大，它们的零点的位置，以及它们在特定数字（特殊值）处的值。这些问题的答案或预期答案具有深刻的算术意义，例如Birch和Swinnerton- Dyer猜想（另一个千年问题！）及其推广。我们研究计划的两个长远目标是研究L函数的特殊值，以及与L函数相关的统计研究，特别是它们在某些点上的不消失。关于L-函数的特殊值的第一个方向是由我们对多变量多项式的马勒测度的关注所驱动的。非零多项式的（对数）马勒测度被定义为一定的复积分，并且已被发现产生具有数论意义的函数的特殊值，如l函数。人们期望理解这些公式将产生更多关于特殊值性质的信息。我们一直致力于这些公式的发现、证明和理解，目的是获得对Beilinson和Bloch关于l函数特殊值的深刻猜想的知识。我们也在研究与离散动力系统相关的动态马勒测度。我们研究计划的另一个主要方向是围绕l函数的统计。通过Montgomery，然后是Katz和Sarnak的工作，很自然地研究L函数族（具有共同算术结构的L函数集），并期望该族的行为将提供有关其单个成员的信息。我们一直在关注L函数的算术统计的各个方面，特别是三次L函数。这个理论对于二次l函数很好理解，但是对于三次的情况就不太了解了，因为它要困难得多。我们一直在研究这些家庭中价值的分布，特别是不消失的结果。

L--functions play a central role in number theory as many of the deepest questions in the area revolve around them. They encode properties of arithmetic objects, such as the prime numbers. Euclid proved that there are infinitely many prime numbers 2300 years ago, and Gauss conjectured an asymptotic formula for the number of primes 200 years ago. This asymptotic formula was eventually proven by Hadamard and de la Vallée Poussin in 1896 and became known as the Prime Number Theorem. However, obtaining a formula as precise as Gauss conjectured is still an open problem, as it depends on the Riemann Hypothesis (RH), one of the seven Millennium Problems from the Clay Mathematics Institute with a prize of one million dollars. RH is a statement about the zeroes of Riemann zeta function, which is the simplest possible L-function. Many directions arise when considering more general L-functions. The most fundamental questions in this topic are concerned with how large L-functions can be, with the location of their zeroes, and with the values they take at particular numbers (special values). The answers or expected answers to such questions have deep arithmetic significance, such as the Birch and Swinnerton--Dyer conjecture (another Millennium Problem!) and its generalizations. The two far--reaching goals of our research program are concerned with the study of special values of L--functions, and the study of statistics associated to L-functions, particularly of their non-vanishing at certain points. The first direction concerning special values of L--functions has been driven by our focus on Mahler measure of multivariable polynomials. The (logarithmic) Mahler measure of a non-zero polynomial is defined as certain complex integral, and has been found to yield special values of functions of number theoretic significance such as L-functions. One expects that understanding these formulas will yield more information about the nature of the special values. We have been working in the discovery, proof, and understanding of such formulas with the goal of gaining knowledge towards deep conjectures of Beilinson and Bloch on special values of L-functions. We are also studying the dynamical Mahler measure, associated to discrete dynamical systems. The other main direction for our research program revolves around statistics of L-functions. By the work of Montgomery, and then Katz and Sarnak, it is natural to study families of L--functions (sets of L--functions that share a common arithmetic structure) with the expectation that the behavior of the family will provide information about its individual members. We have been focusing on various aspects of the arithmetic statistics of L--functions, in particular for cubic L-functions. The theory is well understood for quadratic L-functions, but much less in known for the cubic case, as it is much more difficult. We have been working on the distribution of values in such families, and in particular non-vanishing results.