L-functions and equidistribution

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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.