{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/132565"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/132565","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Topics in analytic number theory","abstract":"This thesis, consisting of two chapters, proves several new theorems concerning L-functions and the distribution of primes. In the first chapter, we establish the first explicit form of the Vinogradov–Korobov zero-free region for Dirichlet L-functions. In the second chapter, we generalize recent work on large gaps between primes to imaginary quadratic fields. Suppose K is an imaginary quadratic field, and let N_K denote the field norm on O_K. For x₀ in O_K and r > 0, let (x₀, r) = { x in O_K : |N_K(x − x₀)| < r }. Define G_K(X) = max { r > 0 : there exists x₀ in O_K such that |N_K(x₀)| ≤ X and B(x₀, r) contains no primes }. We show that G_K(X) is at least c_K (log X) (log₂ X · log₄ X) / log₃ X for some constant c_K > 0 depending only on K.","abstract_html":"This thesis, consisting of two chapters, proves several new theorems concerning L-functions and the distribution of primes. In the first chapter, we establish the first explicit form of the Vinogradov–Korobov zero-free region for Dirichlet L-functions. In the second chapter, we generalize recent work on large gaps between primes to imaginary quadratic fields. Suppose K is an imaginary quadratic field, and let N_K denote the field norm on O_K. For x₀ in O_K and r &gt; 0, let (x₀, r) = { x in O_K : |N_K(x − x₀)| &lt; r }. Define G_K(X) = max { r &gt; 0 : there exists x₀ in O_K such that |N_K(x₀)| ≤ X and B(x₀, r) contains no primes }. We show that G_K(X) is at least c_K (log X) (log₂ X · log₄ X) / log₃ X for some constant c_K &gt; 0 depending only on K.","abstract_has_math":false,"creators":["Khale, Tanmay"],"institution":"University of Illinois Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ford, Kevin B","Thorner, Jesse A","Zaharescu, Alexandru","Gabdullin, Mikhail"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-12","date_published":"2025-12","updated_at":"2026-07-22T22:25:07Z","subjects":["analytic number theory","prime number theory","Dirichlet L-functions","L-functions","zero-free regions","primes in arithmetic progressions","prime gaps","bounded gaps between primes","large gaps between primes","prime k-tuples","Gaussian integers","Gaussian primes"],"languages":["en"],"rights":["Copyright 2025 Tanmay Khale"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/132565","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ford, Kevin B","Thorner, Jesse A","Zaharescu, Alexandru","Gabdullin, Mikhail"]},{"key":"dc:creator","label":"Author","values":["Khale, Tanmay"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-12","2025-12-05"]},{"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 Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["analytic number theory","prime number theory","Dirichlet L-functions","L-functions","zero-free regions","primes in arithmetic progressions","prime gaps","bounded gaps between primes","large gaps between primes","prime k-tuples","Gaussian integers","Gaussian primes"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2025 Tanmay Khale"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/132565"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis, consisting of two chapters, proves several new theorems concerning L-functions and the distribution of primes. In the first chapter, we establish the first explicit form of the Vinogradov–Korobov zero-free region for Dirichlet L-functions. In the second chapter, we generalize recent work on large gaps between primes to imaginary quadratic fields. Suppose K is an imaginary quadratic field, and let N_K denote the field norm on O_K. For x₀ in O_K and r > 0, let (x₀, r) = { x in O_K : |N_K(x − x₀)| < r }. Define G_K(X) = max { r > 0 : there exists x₀ in O_K such that |N_K(x₀)| ≤ X and B(x₀, r) contains no primes }. We show that G_K(X) is at least c_K (log X) (log₂ X · log₄ X) / log₃ X for some constant c_K > 0 depending only on K.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2026-02-19 without embargo terms","The student, Tanmay Khale, accepted the attached license on 2025-12-03 at 10:19.","The student, Tanmay Khale, submitted this Dissertation for approval on 2025-12-03 at 12:54.","This Dissertation was approved for publication on 2025-12-05 at 10:40.","DSpace SAF Submission Ingestion Package generated from Vireo submission #23040 on 2026-02-19 at 18:26:32"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Topics in analytic number theory"]}]}],"canonical_facts":{"dc:contributor":["Ford, Kevin B","Thorner, Jesse A","Zaharescu, Alexandru","Gabdullin, Mikhail"],"dc:creator":["Khale, Tanmay"],"dc:date":["2025-12","2025-12-05"],"dc:description":["This thesis, consisting of two chapters, proves several new theorems concerning L-functions and the distribution of primes. In the first chapter, we establish the first explicit form of the Vinogradov–Korobov zero-free region for Dirichlet L-functions. In the second chapter, we generalize recent work on large gaps between primes to imaginary quadratic fields. Suppose K is an imaginary quadratic field, and let N_K denote the field norm on O_K. For x₀ in O_K and r > 0, let (x₀, r) = { x in O_K : |N_K(x − x₀)| < r }. Define G_K(X) = max { r > 0 : there exists x₀ in O_K such that |N_K(x₀)| ≤ X and B(x₀, r) contains no primes }. We show that G_K(X) is at least c_K (log X) (log₂ X · log₄ X) / log₃ X for some constant c_K > 0 depending only on K.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2026-02-19 without embargo terms","The student, Tanmay Khale, accepted the attached license on 2025-12-03 at 10:19.","The student, Tanmay Khale, submitted this Dissertation for approval on 2025-12-03 at 12:54.","This Dissertation was approved for publication on 2025-12-05 at 10:40.","DSpace SAF Submission Ingestion Package generated from Vireo submission #23040 on 2026-02-19 at 18:26:32"],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/132565"],"dc:language":["en"],"dc:rights":["Copyright 2025 Tanmay Khale"],"dc:subject":["analytic number theory","prime number theory","Dirichlet L-functions","L-functions","zero-free regions","primes in arithmetic progressions","prime gaps","bounded gaps between primes","large gaps between primes","prime k-tuples","Gaussian integers","Gaussian primes"],"dc:title":["Topics in analytic number theory"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:07Z"}