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成果是著名的素数定理，该定理是通过利用黎曼 zeta 函数的素数和零点之间的显着联系而建立的。黎曼 zeta 函数是 L 函数的一个示例。我的研究计划涉及这些思想的概括，这些思想适用于广泛的算术设置和其他 L 函数。我关注的是数论函数在算术均分的微妙阈值附近如何表现，以及猜想的真理在多大程度上平均成立。我根据自然算术结构（即切博塔列夫密度定理）施加的更一般代数约束来研究素数的分布。根据概率数论的模型，一旦它们的大小超过自然阈值，这些素数就有望在这些约束下均匀分布。尚未得到证实的大黎曼假设 (GRH) 预测的估计值接近猜想事实。这个猜想阈值极其微妙，并且与数论中的许多其他猜想密切相关。随后的进步导致了数域、算术统计、二元二次形式、类群中的扭转、椭圆曲线和自同构形式的应用。 我还提出了与这些素数相对应的 L 函数的理论，以及它们与这些算术上更复杂的设置中素数均匀分布阈值的关系。这包括对其零点的水平和垂直分布的详细了解。一项核心计划目标是产生可替代 GRH 的统计结果。例如，可以量化 L 函数族中 L 函数的比例可以满足所需的分析属性。不幸的是，许多现有的分析进展属于低阶分析进展，对于更一般代数情况中出现的高阶 L 函数不可用或不充分。对于实际应用来说，结果往往不够统一，并且文献缺乏足够的计算数据来提出有根据的猜想。我计划构建高效的计算工具并进一步开发技术（例如幂和法）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，该框架对高阶 L 函数进行建模并分析该模型的预期属性。这是一个双重视角，我假设素数均匀分布并研究分析结果。我计划利用概率数论和乘法函数论的技术和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 字符的部分和的取消。我已经在几部即将出版的联合和独立著作中启动了这项研究。