University of Illinois Urbana-Champaign

Zeros and moments of L-functions and applications

Liu, Di

Content Files
LIU-DISSERTATION-2024.pdf

Permalink

https://hdl.handle.net/2142/125601

Description

Title
Zeros and moments of L-functions and applications
Author(s)
Liu, Di
Issue Date
2024-07-11
Director of Research (if dissertation) or Advisor (if thesis)
Zaharescu, Alexandru
Doctoral Committee Chair(s)
Berndt, Bruce
Committee Member(s)
  • Thorner, Jesse
  • Nath, Kunjakanan
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Riemann zeta function, L-functions
Abstract
Several problems regarding the zeros and moments of L-functions are considered in this thesis. We settle two conjectures of Matiyasevich in Chapter 1 on a certain approximation of the Riemann zeta function by a finite and symmetrized version of its Euler product. In Chapter 2 we work on the zeros of Dirichlet L-functions. Ford and Zaharescu in [23] show that the distribution of the zeros of the Riemann zeta function exhibits certain periodic distribution after proper normalization. Here we show that a similar phenomenon appears in the case of Dirichlet L-functions as well. In Chapter 3, this result is extended to GL2 L-functions. We show that the distribution is more complicated and related to the Sato–Tate conjecture. In addition, an analogue of the classic zero density estimate is proved for these L-functions, which enables us to prove a central limit theorem for their logarithms on the critical line. The following Chapter 4 is on the Laguerre–Pólya inequalities for Dirichlet L-functions. These inequalities have been shown to be a necessary condition for the Generalized Riemann Hypothesis. We establish that such inequalities hold true for a positive proportion of a certain family of Dirichlet L-functions. In the final Chapter 5 we compute an estimate for the mollified and shifted fourth moment of Dirichlet L-functions along the critical line. Such estimates have numerous uses in analytic number theory. As an example, the result in Chapter 4 is a direct consequence.
Graduation Semester
2024-08
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/125601
Copyright and License Information
Copyright 2024 Di Liu

Owning Collections

Graduate Dissertations and Theses at Illinois PRIMARY
Graduate Theses and Dissertations at Illinois