Several aspects of L-functions

L-函数的几个方面

• 批准号: RGPIN-2022-03651
• 负责人: Lalin, Matilde
• 金额: $ 2.26万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759374
• 关键词: Several; aspects; functions

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.

L-函数在数论中发挥着核心作用，因为该领域的许多最深奥的问题都围绕着它们进行编码，例如欧几里得在 2300 年前就证明了素数的数量是无限的。高斯在200年前猜想了素数个数的渐近公式，这个渐近公式最终被Hadamard和de la Vallée Poussin于1896年证明。并被称为素数定理。然而，获得像高斯猜想一样精确的公式仍然是一个悬而未决的问题，因为它取决于克莱曼数学研究所获奖的七大千年难题之一的黎曼猜想（RH）。 RH 是关于黎曼 zeta 函数零点的陈述，这是最简单的 L 函数，在考虑更一般的 L 函数时会出现许多方向。本主题涉及 L 函数可以有多大、其零点的位置以及它们在特定数字（特殊值）处取的值。此类问题的答案或预期答案具有深刻的算术意义，例如。作为伯奇和斯温纳顿-戴尔猜想（另一个千年问题！）及其推广，我们研究计划的两个深远目标涉及 L 函数的特殊值的研究和相关统计关于 L 函数的特殊值的第一个方向是由我们对多元多项式的马勒测度的关注所驱动的。零多项式被定义为某种复积分，并且已被发现可以产生具有数论意义的函数的特殊值，例如 L 函数。人们期望理解这些公式将产生有关特殊值的性质的更多信息。我们一直致力于发现、证明和理解这些公式，目的是获得 Beilinson 和 Bloch 关于 L 函数特殊值的深层猜想的知识。我们还在研究与离散相关的动态马勒测度。动力系统 我们研究项目的另一个主要方向围绕着 L 函数的统计，通过 Montgomery、Katz 和 Sarnak 的工作，研究 L 函数族（L 函数的集合）是很自然的。共享共同的算术结构），期望族的行为能够提供有关其个体成员的信息。我们一直关注 L 函数算术统计的各个方面，特别是三次 L 函数的理论。对于二次 L 函数很好理解，但对于三次函数则知之甚少，因为我们一直致力于研究此类族中值的分布，特别是非消失结果。