{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/129422"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/129422","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Combinatorial methods in number theory: Sieve theory and special functions","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms","abstract_has_math":false,"creators":["Xie, Likun"],"institution":"University of Illinois Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Zaharescu, Alexandru","Berndt, Bruce Carl","Reznick, Bruce","Nath, Kunjakanan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-04-24","date_published":"2025-04-24","updated_at":"2026-07-22T22:25:05Z","subjects":["Mctintosh conjecture","almost primes","vector sieves","elliptic curves","twin primes"],"languages":["en","eng"],"rights":["Copyright 2025 Likun Xie"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/129422","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zaharescu, Alexandru","Berndt, Bruce Carl","Reznick, Bruce","Nath, Kunjakanan"]},{"key":"dc:creator","label":"Author","values":["Xie, Likun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-04-24","2025-05"]},{"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":["Mctintosh conjecture","almost primes","vector sieves","elliptic curves","twin primes"]}]},{"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 2025 Likun Xie"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/129422"]}]},{"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 2025-10-19 without embargo terms","The student, Likun Xie, accepted the attached license on 2025-04-22 at 11:09.","The student, Likun Xie, submitted this Dissertation for approval on 2025-04-22 at 11:28.","This Dissertation was approved for publication on 2025-04-24 at 07:02.","DSpace SAF Submission Ingestion Package generated from Vireo submission #21886 on 2025-10-19 at 18:18:36","This dissertation explores various problems in analytic number theory, using a blend of sieve-theoretic methods, algebraic, and combinatorial techniques. The work is split into two main parts. First, we study sieve methods and applications. We investigate counting results for primes and almost primes with additional constraints, such as those related to orders of elliptic curves modulo prime powers, primes p with large power factors in p - b, and primes of the form p = 1 + m^2 + n^2 such that p + 2 is an almost prime. These problems are approached using vector sieve, linear sieve, semi-linear sieve, and other advanced sieve frameworks, often combined with analytic estimates like the Bombieri–Vinogradov theorem in various contexts. Second, we address Franel integrals and arithmetic properties. We prove McIntosh’s conjecture on certain multidimensional Franel integrals involving Bernoulli polynomials and higher-dimensional analogs. By unifying these two directions—sieve applications and Franel-type integrals—we illustrate the power of combinatorial-analytic techniques in tackling problems within number theory. In particular, we show how seemingly disparate areas (twin primes and almost primes, the orders of elliptic curves over finite fields, and integrals involving Bernoulli functions) can reinforce one another."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Combinatorial methods in number theory: Sieve theory and special functions"]}]}],"canonical_facts":{"dc:contributor":["Zaharescu, Alexandru","Berndt, Bruce Carl","Reznick, Bruce","Nath, Kunjakanan"],"dc:creator":["Xie, Likun"],"dc:date":["2025-04-24","2025-05"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms","The student, Likun Xie, accepted the attached license on 2025-04-22 at 11:09.","The student, Likun Xie, submitted this Dissertation for approval on 2025-04-22 at 11:28.","This Dissertation was approved for publication on 2025-04-24 at 07:02.","DSpace SAF Submission Ingestion Package generated from Vireo submission #21886 on 2025-10-19 at 18:18:36","This dissertation explores various problems in analytic number theory, using a blend of sieve-theoretic methods, algebraic, and combinatorial techniques. The work is split into two main parts. First, we study sieve methods and applications. We investigate counting results for primes and almost primes with additional constraints, such as those related to orders of elliptic curves modulo prime powers, primes p with large power factors in p - b, and primes of the form p = 1 + m^2 + n^2 such that p + 2 is an almost prime. These problems are approached using vector sieve, linear sieve, semi-linear sieve, and other advanced sieve frameworks, often combined with analytic estimates like the Bombieri–Vinogradov theorem in various contexts. Second, we address Franel integrals and arithmetic properties. We prove McIntosh’s conjecture on certain multidimensional Franel integrals involving Bernoulli polynomials and higher-dimensional analogs. By unifying these two directions—sieve applications and Franel-type integrals—we illustrate the power of combinatorial-analytic techniques in tackling problems within number theory. In particular, we show how seemingly disparate areas (twin primes and almost primes, the orders of elliptic curves over finite fields, and integrals involving Bernoulli functions) can reinforce one another."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/129422"],"dc:language":["en","eng"],"dc:rights":["Copyright 2025 Likun Xie"],"dc:subject":["Mctintosh conjecture","almost primes","vector sieves","elliptic curves","twin primes"],"dc:title":["Combinatorial methods in number theory: Sieve theory and special functions"],"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":"2026-07-22T22:25:05Z"}