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oai:www.ideals.illinois.edu:2142/125601 2025-02-06 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-02-04 without embargo terms The student, Di Liu, accepted the attached license on 2024-07-09 at 21:22. The student, Di Liu, submitted this Dissertation for approval on 2024-07-09 at 21:26. This Dissertation was approved for publication on 2024-07-11 at 11:21. DSpace SAF Submission Ingestion Package generated from Vireo submission #21030 on 2025-02-04 at 21:04:51 Zeros and moments of L-functions and applications Liu, Di 2024-07-11 Zaharescu, Alexandru Berndt, Bruce Thorner, Jesse Nath, Kunjakanan Riemann Zeta Function L-functions eng 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. 2024-08 Thesis https://hdl.handle.net/2142/125601 Copyright 2024 Di Liu Mathematics Mathematics University of Illinois at Urbana-Champaign Ph.D. Dissertation