L-functions and equidistribution

L 函数和均匀分布

• 批准号: RGPIN-2022-04982
• 负责人: Zaman, Asif
• 金额: $ 1.68万
• 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: 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=758705
• 关键词: functions; equidistribution

A central result in analytic number theory is the celebrated Prime Number Theorem, which was established by exploiting remarkable connections between primes and zeros of Riemann's zeta function. The Riemann zeta function is an example of a L-function. My research program involves a generalization of these ideas that apply to a wide range of arithmetic settings and other L-functions. I focus on how number theoretic functions behave near the delicate threshold of arithmetic equidistribution and to what extent the conjectural truth holds on average. I study the distribution of primes with respect to more general algebraic constraints imposed by natural arithmetic structures, namely the Chebotarev density theorem. Based on models from probabilistic number theory, such primes are expected to equidistribute with respect to these constraints once their size surpasses a natural threshold. The still unproven Grand Riemann Hypothesis (GRH) predicts an estimate close to the conjectural truth. This conjectural threshold is extremely delicate and deeply connected to many other conjectures in number theory. Advances subsequently lead to applications for number fields, arithmetic statistics, binary quadratic forms, torsion in class groups, elliptic curves, and automorphic forms. I also advance the theory of the L-functions corresponding to these primes and their relationship to the threshold for equidistribution of primes in these more arithmetically complicated settings. This includes a detailed understanding of the horizontal and vertical distribution of their zeros. One core program objective is to produce results which serve as statistical substitutes for the GRH. For example, one may quantify what proportion of L-functions in a family of L-functions may satisfy a desirable analytic property. Unfortunately, many existing analytic advances pertain to those of low degree and are unavailable or insufficient for the high degree L-functions that arise in more general algebraic situations. Results are often not sufficiently uniform for practical applications and the literature lacks enough computational data to formulate well-founded conjectures. I plan to construct efficient computational tools and further develop techniques, such as the power sum method, to study these L-functions. I am actively developing theoretical frameworks based on random multiplicative functions that model high degree L-functions and analyze the expected properties of this model. This is a dual perspective where I instead assume primes equidistribute and investigate the analytic consequences. I plan to leverage the technology and recent developments from probabilistic number theory and multiplicative function theory to study these frameworks and its applications to general arithmetic settings, such as cancellation in partial sums of irreducible Artin characters. I have already initiated this study in several forthcoming joint and independent works.

解析数论的一个核心结果是著名的素数定理，它是通过利用黎曼Zeta函数的素数和零点之间的显著联系而建立的。Riemann Zeta函数是L函数的一个例子。我的研究项目涉及到这些思想的概括，这些思想适用于广泛的算术设置和其他L函数。我的重点是数论函数在算术均匀分布这一微妙门槛附近的表现，以及猜想真理在多大程度上平均成立。我研究了关于自然算术结构施加的更一般的代数约束的素数的分布，即切博塔雷夫密度定理。根据概率数论的模型，一旦这些素数的大小超过自然阈值，它们就有望相对于这些约束均匀分布。仍未得到证实的大黎曼假说(GRH)预测了一个接近猜想真相的估计。这个猜想的门槛非常微妙，与数论中的许多其他猜想有很深的联系。这些进展随后导致了数域、算术统计、二进制二次型、类群中的扭转、椭圆曲线和自同构形式的应用。还提出了与这些素数对应的L函数的理论，以及它们与素数在这些更复杂的设置下的均匀分布门限的关系。这包括详细了解它们的零点的水平和垂直分布。方案的一个核心目标是产生可作为GRH的统计替代品的结果。例如，可以量化L函数在L函数族中的多大比例可以满足理想的解析性质。遗憾的是，许多现有的分析进展都是低次的，对于更一般的代数情形中出现的高次L函数，它们是不可用的或不充分的。对于实际应用，结果往往不够统一，文献缺乏足够的计算数据来提出有充分根据的猜想。我计划构建高效的计算工具，并进一步发展技术，如幂和方法，来研究这些L函数。我正在积极开发基于随机乘法函数的理论框架，对高次L函数进行建模，并分析该模型的预期性质。这是一个双重视角，我假设素数均匀分布，并研究分析结果。我计划利用这项技术和概率数论和乘法函数理论的最新发展来研究这些框架及其在一般算术设置中的应用，例如不可约Artin字符的部分和的消除。我已经在即将到来的几部联合和独立的作品中开始了这项研究。