{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71198"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71198","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Explicit Estimates for Functions of Primes in Arithmetic Progressions","abstract":"This thesis is concerned with essentially three topics: Explicit zero-free regions for Dirichlet L-functions, numerical estimates for the error term in the prime number theorem for arithmetic progressions, and Waring's problem for cubes.","abstract_html":"This thesis is concerned with essentially three topics: Explicit zero-free regions for Dirichlet L-functions, numerical estimates for the error term in the prime number theorem for arithmetic progressions, and Waring's problem for cubes.","abstract_has_math":false,"creators":["Mccurley, Kevin Snow"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:01Z","date_published":"2014-12-16T06:18:01Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8203527"],"render_values":[{"text":"(UMI)AAI8203527","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71198","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Mccurley, Kevin Snow"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:01Z","10000-01-01","1981"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"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":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71198","(UMI)AAI8203527"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis is concerned with essentially three topics: Explicit zero-free regions for Dirichlet L-functions, numerical estimates for the error term in the prime number theorem for arithmetic progressions, and Waring's problem for cubes.","In chapter 1 the following result is proved: Among the (phi)(k) characters (chi) modulo k there is at most one character for which the Dirichlet L-function L(s,(chi)) has a zero (rho) = (beta) + i(gamma) with (beta) > 1 - 1/(Rlogq), where R = 9.645908801, and q = max{k, k(VBAR)(gamma)(VBAR), 30}. If such a zero exists it is a real zero of an L-function formed with a real non-principal character. Several methods of proof are discussed for showing that a given modulus k does not admit an exceptional zero.","In chapter 2 explicit numerical values are given for constants C(,1) and C(,2) with the property that","(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)","where (chi) is a primitive character modulo k, and N(T,(chi)) counts the number of zeros of L(s,(chi)) with 0 < (beta) < 1 and (VBAR)(gamma)(VBAR) (LESSTHEQ) T.","The object of chapter 3 is the estimation of the Chebyshev functions (theta)(x;k,l) and (psi)(x;k,l). For various values of (epsilon), tables and c and b are given for which it can be asserted that","provided that (k,l) = 1, x (GREATERTHEQ) exp(clog('2)k), k (GREATERTHEQ) 10('b), and the modulus k does not admit an exceptional zero. The method used in the proof is similar to that used by Rosser and Schoenfeld in the case k = 1, where an integral average of the function (psi)(x;k,l) is expressed in an explicit formula involving the zeros of Dirichlet L-functions. The explicit formula can then be estimated directly with the use of results from chapters 1 and 2.","Chapter 4 considers the case k = 3 in more detail. In this case the results of chapter 3 can be sharpened by making use of extensive numerical information concerning the zeros of the two Dirichlet L-functions modulo 3.","Waring's problem for cubes is the topic of chapter 5. It is proved that every integer exceeding exp(1.1 x 10('6)) is a sum of seven non-negative integral cubes. Previous proofs of the seven cube theorem were ineffective due to the use of the Siegel-Walfisz theorem. Numerical evidence is presented for the conjecture that every integer exceeding 1290740 is a sum of five non-negative integral cubes.","Made available in DSpace on 2014-12-16T06:18:01Z (GMT). No. of bitstreams: 1 8203527.pdf: 3225365 bytes, checksum: a0759f94faae8280328006254f73a91a (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 71364 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","134 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."]},{"key":"dc:title","label":"Title","values":["Explicit Estimates for Functions of Primes in Arithmetic Progressions"]}]}],"canonical_facts":{"dc:creator":["Mccurley, Kevin Snow"],"dc:date":["2014-12-16T06:18:01Z","10000-01-01","1981"],"dc:description":["This thesis is concerned with essentially three topics: Explicit zero-free regions for Dirichlet L-functions, numerical estimates for the error term in the prime number theorem for arithmetic progressions, and Waring's problem for cubes.","In chapter 1 the following result is proved: Among the (phi)(k) characters (chi) modulo k there is at most one character for which the Dirichlet L-function L(s,(chi)) has a zero (rho) = (beta) + i(gamma) with (beta) > 1 - 1/(Rlogq), where R = 9.645908801, and q = max{k, k(VBAR)(gamma)(VBAR), 30}. If such a zero exists it is a real zero of an L-function formed with a real non-principal character. Several methods of proof are discussed for showing that a given modulus k does not admit an exceptional zero.","In chapter 2 explicit numerical values are given for constants C(,1) and C(,2) with the property that","(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)","where (chi) is a primitive character modulo k, and N(T,(chi)) counts the number of zeros of L(s,(chi)) with 0 < (beta) < 1 and (VBAR)(gamma)(VBAR) (LESSTHEQ) T.","The object of chapter 3 is the estimation of the Chebyshev functions (theta)(x;k,l) and (psi)(x;k,l). For various values of (epsilon), tables and c and b are given for which it can be asserted that","provided that (k,l) = 1, x (GREATERTHEQ) exp(clog('2)k), k (GREATERTHEQ) 10('b), and the modulus k does not admit an exceptional zero. The method used in the proof is similar to that used by Rosser and Schoenfeld in the case k = 1, where an integral average of the function (psi)(x;k,l) is expressed in an explicit formula involving the zeros of Dirichlet L-functions. The explicit formula can then be estimated directly with the use of results from chapters 1 and 2.","Chapter 4 considers the case k = 3 in more detail. In this case the results of chapter 3 can be sharpened by making use of extensive numerical information concerning the zeros of the two Dirichlet L-functions modulo 3.","Waring's problem for cubes is the topic of chapter 5. It is proved that every integer exceeding exp(1.1 x 10('6)) is a sum of seven non-negative integral cubes. Previous proofs of the seven cube theorem were ineffective due to the use of the Siegel-Walfisz theorem. Numerical evidence is presented for the conjecture that every integer exceeding 1290740 is a sum of five non-negative integral cubes.","Made available in DSpace on 2014-12-16T06:18:01Z (GMT). No. of bitstreams: 1 8203527.pdf: 3225365 bytes, checksum: a0759f94faae8280328006254f73a91a (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 71364 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","134 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."],"dc:identifier":["http://hdl.handle.net/2142/71198","(UMI)AAI8203527"],"dc:subject":["Mathematics"],"dc:title":["Explicit Estimates for Functions of Primes in Arithmetic Progressions"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}