Zeros and moments of L-functions and applications

Liu, Di

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

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

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