{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19706"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19706","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A new upper bound in the linear sieve and its applications","abstract":"In Chapter I we shall prove a new upper bound in the linear sieve. Our purpose in Chapter II is to explain our method in greater detail than was done in Chapter I. Let x be a large number. We consider $\\pi\\sb2$(x)--the number of prime twins not exceeding x. Using the new upper bound in the linear sieve from Chapter I, we shall prove that$$\\rm\\pi\\sb2({x}) 0$ and x $\\geq$ x$\\sb0(\\epsilon),$ where$$\\rm H = 2{\\prod\\limits\\sb{p>2}}\\left(1-{1\\over(p-1)\\sp2}\\right).$$In the Appendix, various computations cited in the text are given in detail.","abstract_html":"In Chapter I we shall prove a new upper bound in the linear sieve. Our purpose in Chapter II is to explain our method in greater detail than was done in Chapter I. Let x be a large number. We consider <span class=\"etd-inline-math\">&pi;\\sb2</span>(x)--the number of prime twins not exceeding x. Using the new upper bound in the linear sieve from Chapter I, we shall prove that$<span class=\"etd-inline-math\">\\rm&pi;\\sb2({x}) 0</span> and x $\\geq$ x<span class=\"etd-inline-math\">\\sb0(&epsilon;),</span> where$$\\rm H = 2{\\prod\\limits\\sb{p&gt;2}}\\left(1-{1\\over(p-1)\\sp2}\\right).$$In the Appendix, various computations cited in the text are given in detail.","abstract_has_math":true,"creators":["Lou, Shituo"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Diamond, Harold G."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:15:58Z","date_published":"2011-05-07T12:15:58Z","updated_at":"2026-07-22T22:25:14Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Lou, Shituo"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9021722","(UMI)AAI9021722"],"render_values":[{"text":"AAI9021722","href":null,"code":true},{"text":"(UMI)AAI9021722","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19706","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Diamond, Harold G."]},{"key":"dc:creator","label":"Author","values":["Lou, Shituo"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:15:58Z","10000-01-01","1990"]},{"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":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 Lou, Shituo"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9021722","(UMI)AAI9021722","http://hdl.handle.net/2142/19706"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In Chapter I we shall prove a new upper bound in the linear sieve. Our purpose in Chapter II is to explain our method in greater detail than was done in Chapter I. Let x be a large number. We consider $\\pi\\sb2$(x)--the number of prime twins not exceeding x. Using the new upper bound in the linear sieve from Chapter I, we shall prove that$$\\rm\\pi\\sb2({x}) 0$ and x $\\geq$ x$\\sb0(\\epsilon),$ where$$\\rm H = 2{\\prod\\limits\\sb{p>2}}\\left(1-{1\\over(p-1)\\sp2}\\right).$$In the Appendix, various computations cited in the text are given in detail.","Made available in DSpace on 2011-05-07T12:15:58Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9021722.pdf: 1658645 bytes, checksum: 7176bdd87e5f392556120f02f21d150f (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:38:53Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:16:16-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["A new upper bound in the linear sieve and its applications"]}]}],"canonical_facts":{"dc:contributor":["Diamond, Harold G."],"dc:creator":["Lou, Shituo"],"dc:date":["2011-05-07T12:15:58Z","10000-01-01","1990"],"dc:description":["In Chapter I we shall prove a new upper bound in the linear sieve. Our purpose in Chapter II is to explain our method in greater detail than was done in Chapter I. Let x be a large number. We consider $\\pi\\sb2$(x)--the number of prime twins not exceeding x. Using the new upper bound in the linear sieve from Chapter I, we shall prove that$$\\rm\\pi\\sb2({x}) 0$ and x $\\geq$ x$\\sb0(\\epsilon),$ where$$\\rm H = 2{\\prod\\limits\\sb{p>2}}\\left(1-{1\\over(p-1)\\sp2}\\right).$$In the Appendix, various computations cited in the text are given in detail.","Made available in DSpace on 2011-05-07T12:15:58Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9021722.pdf: 1658645 bytes, checksum: 7176bdd87e5f392556120f02f21d150f (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:38:53Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:16:16-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9021722","(UMI)AAI9021722","http://hdl.handle.net/2142/19706"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Lou, Shituo"],"dc:subject":["Mathematics"],"dc:title":["A new upper bound in the linear sieve and its applications"],"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":"2026-07-22T22:25:14Z"}