L-functions and equidistribution

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ζ函数中素数和零之间的显著联系建立起来的。黎曼函数是l函数的一个例子。我的研究计划涉及到这些思想的推广，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匀分布的微妙阈值附近是如何表现的，以及猜想的真理在多大程度上保持平均。我研究质数在自然算术结构所施加的更一般代数约束下的分布，即Chebotarev密度定理。根据概率数论的模型，一旦质数的大小超过自然阈值，这些质数就会在这些约束条件下均匀分布。尚未被证明的大黎曼假设（GRH）预测了一个接近推测真理的估计。这个猜想的阈值非常微妙，并且与数论中的许多其他猜想密切相关。随后的进展导致了数域、算术统计、二元二次型、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，以及它们在这些更复杂的算术设置中与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。该项目的一个核心目标是产生可作为GRH统计替代品的结果。例如，可以量化l -函数族中l -函数满足理想解析性质的比例。不幸的是，许多现有的解析进展都是关于低次的，对于在更一般的代数情况下出现的高次l函数是不可用的或不够的。对于实际应用，结果往往不够统一，而且文献缺乏足够的计算数据来制定有充分根据的推测。我计划构建高效的计算工具，并进一步开发诸如幂和法等技术来研究这些l -函数。我正在积极开发基于随机乘法函数的理论框架，对高阶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，我假设质数是等分布的，并研究分析结果。我计划利用概率数论和乘法函数理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字符部分和的消去。我已经在几部即将出版的联合和独立作品中开始了这项研究。

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.