{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/108094"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/108094","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Analytic and arithmetic applications of half integral weight automorphic forms","abstract":"In this thesis we explore both analytic and arithmetic applications of half integral weight modular forms. In the first chapter we are motivated by a conjecture of Hoffstein (2011) that asserts that the L-series attached to a half integral weight modular form satisfies a Lindelof hypothesis. Using spectral and diophantine techniques, the first chapter establishes a twisted second moment of L-series attached to such forms with a power saving error term. The second chapter of this thesis uses the spectral theory of half integral weight forms to establish a power saving in a formula of Andrews for the coefficients of Ramanujan's famous third order mock theta function. We also prove a conjecture of Andrews (1966) relating to the convergence of said formula. The third chapter uses spectral techniques and half integral weight Hecke theory to get bounds on sums of half integral weight Kloosterman sums. This is inspired by Sarnak and Tsimerman's treatment of weight zero Kloosterman sums in the context of the Linnik--Selberg conjecture. Each chapter is completely self contained and independent of the others. They were originally written as separate manuscripts for publication.","abstract_html":"In this thesis we explore both analytic and arithmetic applications of half integral weight modular forms. In the first chapter we are motivated by a conjecture of Hoffstein (2011) that asserts that the L-series attached to a half integral weight modular form satisfies a Lindelof hypothesis. Using spectral and diophantine techniques, the first chapter establishes a twisted second moment of L-series attached to such forms with a power saving error term. The second chapter of this thesis uses the spectral theory of half integral weight forms to establish a power saving in a formula of Andrews for the coefficients of Ramanujan's famous third order mock theta function. We also prove a conjecture of Andrews (1966) relating to the convergence of said formula. The third chapter uses spectral techniques and half integral weight Hecke theory to get bounds on sums of half integral weight Kloosterman sums. This is inspired by Sarnak and Tsimerman's treatment of weight zero Kloosterman sums in the context of the Linnik--Selberg conjecture. Each chapter is completely self contained and independent of the others. They were originally written as separate manuscripts for publication.","abstract_has_math":false,"creators":["Dunn, Alexander"],"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","Ahlgren , Scott","Ford, Kevin","Allen , Patrick"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-08-26T23:51:25Z","date_published":"2020-08-26T23:51:25Z","updated_at":"2026-07-22T22:24:47Z","subjects":["L function, moment, partition, Kloosterman sums, Maass forms"],"languages":["en"],"rights":["Copyright 2020 Alexander Dunn"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/108094","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zaharescu , Alexandru","Ahlgren , Scott","Ford, Kevin","Allen , Patrick"]},{"key":"dc:creator","label":"Author","values":["Dunn, Alexander"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-08-26T23:51:25Z","2022-08-26T23:58:55Z","2020-03-31","2020-05"]},{"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":["L function, moment, partition, Kloosterman sums, Maass forms"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Alexander Dunn"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/108094"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis we explore both analytic and arithmetic applications of half integral weight modular forms. In the first chapter we are motivated by a conjecture of Hoffstein (2011) that asserts that the L-series attached to a half integral weight modular form satisfies a Lindelof hypothesis. Using spectral and diophantine techniques, the first chapter establishes a twisted second moment of L-series attached to such forms with a power saving error term. The second chapter of this thesis uses the spectral theory of half integral weight forms to establish a power saving in a formula of Andrews for the coefficients of Ramanujan's famous third order mock theta function. We also prove a conjecture of Andrews (1966) relating to the convergence of said formula. The third chapter uses spectral techniques and half integral weight Hecke theory to get bounds on sums of half integral weight Kloosterman sums. This is inspired by Sarnak and Tsimerman's treatment of weight zero Kloosterman sums in the context of the Linnik--Selberg conjecture. Each chapter is completely self contained and independent of the others. They were originally written as separate manuscripts for publication.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2022-05-01","The student, Alexander Dunn, accepted the attached license on 2020-03-19 at 15:40.","The student, Alexander Dunn, submitted this Dissertation for approval on 2020-03-19 at 15:50.","This Dissertation was approved for publication on 2020-03-31 at 09:42.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14907 on 2020-08-25 at 17:27:01","Made available in DSpace on 2020-08-26T23:51:25Z (GMT). No. of bitstreams: 2 DUNN-DISSERTATION-2020.pdf: 848688 bytes, checksum: 8ce32ab9e1625a2ad46851ca6133cf77 (MD5) LICENSE.txt: 4211 bytes, checksum: d80aade0f2f894a21fe68bd503c58123 (MD5) Previous issue date: 2020-03-31","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:51:32Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:54:40Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:55:59Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:57:28Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:58:55Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Analytic and arithmetic applications of half integral weight automorphic forms"]}]}],"canonical_facts":{"dc:contributor":["Zaharescu , Alexandru","Ahlgren , Scott","Ford, Kevin","Allen , Patrick"],"dc:creator":["Dunn, Alexander"],"dc:date":["2020-08-26T23:51:25Z","2022-08-26T23:58:55Z","2020-03-31","2020-05"],"dc:description":["In this thesis we explore both analytic and arithmetic applications of half integral weight modular forms. In the first chapter we are motivated by a conjecture of Hoffstein (2011) that asserts that the L-series attached to a half integral weight modular form satisfies a Lindelof hypothesis. Using spectral and diophantine techniques, the first chapter establishes a twisted second moment of L-series attached to such forms with a power saving error term. The second chapter of this thesis uses the spectral theory of half integral weight forms to establish a power saving in a formula of Andrews for the coefficients of Ramanujan's famous third order mock theta function. We also prove a conjecture of Andrews (1966) relating to the convergence of said formula. The third chapter uses spectral techniques and half integral weight Hecke theory to get bounds on sums of half integral weight Kloosterman sums. This is inspired by Sarnak and Tsimerman's treatment of weight zero Kloosterman sums in the context of the Linnik--Selberg conjecture. Each chapter is completely self contained and independent of the others. They were originally written as separate manuscripts for publication.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2022-05-01","The student, Alexander Dunn, accepted the attached license on 2020-03-19 at 15:40.","The student, Alexander Dunn, submitted this Dissertation for approval on 2020-03-19 at 15:50.","This Dissertation was approved for publication on 2020-03-31 at 09:42.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14907 on 2020-08-25 at 17:27:01","Made available in DSpace on 2020-08-26T23:51:25Z (GMT). No. of bitstreams: 2 DUNN-DISSERTATION-2020.pdf: 848688 bytes, checksum: 8ce32ab9e1625a2ad46851ca6133cf77 (MD5) LICENSE.txt: 4211 bytes, checksum: d80aade0f2f894a21fe68bd503c58123 (MD5) Previous issue date: 2020-03-31","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:51:32Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:54:40Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:55:59Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:57:28Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 115703 Lift date: 2022-08-26T23:58:55Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/108094"],"dc:language":["en"],"dc:rights":["Copyright 2020 Alexander Dunn"],"dc:subject":["L function, moment, partition, Kloosterman sums, Maass forms"],"dc:title":["Analytic and arithmetic applications of half integral weight automorphic forms"],"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":"2026-07-22T22:24:47Z"}