Zeros and moments of L-functions and applications

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Liu, Di

Contributors dc:contributor

• Zaharescu, Alexandru
• Berndt, Bruce
• Thorner, Jesse
• Nath, Kunjakanan

• Riemann Zeta Function
• L-functions

Statement dc:rights

• Copyright 2024 Di Liu

Language dc:language

en, eng

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F125601