{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/125601"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/125601","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Zeros and moments of L-functions and applications","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-02-04 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-02-04 without embargo terms","abstract_has_math":false,"creators":["Liu, Di"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Zaharescu, Alexandru","Berndt, Bruce","Thorner, Jesse","Nath, Kunjakanan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-07-11","date_published":"2024-07-11","updated_at":"2026-07-22T22:25:02Z","subjects":["Riemann Zeta Function","L-functions"],"languages":["en","eng"],"rights":["Copyright 2024 Di Liu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/125601","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zaharescu, Alexandru","Berndt, Bruce","Thorner, Jesse","Nath, Kunjakanan"]},{"key":"dc:creator","label":"Author","values":["Liu, Di"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024-07-11","2024-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Riemann Zeta Function","L-functions"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2024 Di Liu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/125601"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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","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."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Zeros and moments of L-functions and applications"]}]}],"canonical_facts":{"dc:contributor":["Zaharescu, Alexandru","Berndt, Bruce","Thorner, Jesse","Nath, Kunjakanan"],"dc:creator":["Liu, Di"],"dc:date":["2024-07-11","2024-08"],"dc:description":["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","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."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/125601"],"dc:language":["en","eng"],"dc:rights":["Copyright 2024 Di Liu"],"dc:subject":["Riemann Zeta Function","L-functions"],"dc:title":["Zeros and moments of L-functions and applications"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:02Z"}