{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86943"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86943","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On Beurling's Theory of Generalized Numbers","abstract":"Following A. Beurling (Acta Math. 68 (1937) 255-291), we consider a set of generalized prime numbers $P=\\{1<p\\sb1\\le p\\sb2\\le\\...\\}$ and the set of generalized integers $N=\\{n\\sb1=1\\le n\\sb2\\le\\...\\}$ generated by P. We let $N(x)$ be the counting function of the set N. In this thesis we give continuous versions of generalized number systems considered by R. S. Hall (Proc. Amer. Math. Soc. 40 (1973) 79-82). We provide an explicit calculation of the associated zeta function, which in turn allows us to obtain an expression for $N(x)$ with several terms rather than just an O-term for the error. We construct examples that illustrate a theorem of W.-B. Zhang (Illinois J. Math. 31 (1987) 645-664) about the mean value of the generalized Mobius function. We construct a continuous number system for which the error term in the integer counting function $N(x)$ oscillates as much as the de la Vallee Poussin error for the Prime Number Theorem. A simplification in a theorem of H. G. Diamond is offered. Finally we estimate the integral of a measure obtained by interpolating the integer counting measure and (essentially) the Lebesgue measure.","abstract_html":"Following A. Beurling (Acta Math. 68 (1937) 255-291), we consider a set of generalized prime numbers $P=\\{1&lt;p\\sb1\\le p\\sb2\\le\\...\\}$ and the set of generalized integers $N=\\{n\\sb1=1\\le n\\sb2\\le\\...\\}$ generated by P. We let $N(x)$ be the counting function of the set N. In this thesis we give continuous versions of generalized number systems considered by R. S. Hall (Proc. Amer. Math. Soc. 40 (1973) 79-82). We provide an explicit calculation of the associated zeta function, which in turn allows us to obtain an expression for $N(x)$ with several terms rather than just an O-term for the error. We construct examples that illustrate a theorem of W.-B. Zhang (Illinois J. Math. 31 (1987) 645-664) about the mean value of the generalized Mobius function. We construct a continuous number system for which the error term in the integer counting function $N(x)$ oscillates as much as the de la Vallee Poussin error for the Prime Number Theorem. A simplification in a theorem of H. G. Diamond is offered. Finally we estimate the integral of a measure obtained by interpolating the integer counting measure and (essentially) the Lebesgue measure.","abstract_has_math":true,"creators":["Balanzario-Gutierrez, Eugenio Pacelli"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Diamond, H.G."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:17Z","date_published":"2015-09-28T15:20:17Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9737042"],"render_values":[{"text":"(MiAaPQ)AAI9737042","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86943","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Diamond, H.G."]},{"key":"dc:creator","label":"Author","values":["Balanzario-Gutierrez, Eugenio Pacelli"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:17Z","10000-01-01","1997"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86943","(MiAaPQ)AAI9737042"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Following A. Beurling (Acta Math. 68 (1937) 255-291), we consider a set of generalized prime numbers $P=\\{1<p\\sb1\\le p\\sb2\\le\\...\\}$ and the set of generalized integers $N=\\{n\\sb1=1\\le n\\sb2\\le\\...\\}$ generated by P. We let $N(x)$ be the counting function of the set N. In this thesis we give continuous versions of generalized number systems considered by R. S. Hall (Proc. Amer. Math. Soc. 40 (1973) 79-82). We provide an explicit calculation of the associated zeta function, which in turn allows us to obtain an expression for $N(x)$ with several terms rather than just an O-term for the error. We construct examples that illustrate a theorem of W.-B. Zhang (Illinois J. Math. 31 (1987) 645-664) about the mean value of the generalized Mobius function. We construct a continuous number system for which the error term in the integer counting function $N(x)$ oscillates as much as the de la Vallee Poussin error for the Prime Number Theorem. A simplification in a theorem of H. G. Diamond is offered. Finally we estimate the integral of a measure obtained by interpolating the integer counting measure and (essentially) the Lebesgue measure.","Made available in DSpace on 2015-09-28T15:20:17Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9737042.pdf: 2431645 bytes, checksum: 36504114a23bca171cac7c06fa3e7138 (MD5) Previous issue date: 1997","Embargo set by: Seth Robbins for item 88224 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","78 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997."]},{"key":"dc:title","label":"Title","values":["On Beurling's Theory of Generalized Numbers"]}]}],"canonical_facts":{"dc:contributor":["Diamond, H.G."],"dc:creator":["Balanzario-Gutierrez, Eugenio Pacelli"],"dc:date":["2015-09-28T15:20:17Z","10000-01-01","1997"],"dc:description":["Following A. Beurling (Acta Math. 68 (1937) 255-291), we consider a set of generalized prime numbers $P=\\{1<p\\sb1\\le p\\sb2\\le\\...\\}$ and the set of generalized integers $N=\\{n\\sb1=1\\le n\\sb2\\le\\...\\}$ generated by P. We let $N(x)$ be the counting function of the set N. In this thesis we give continuous versions of generalized number systems considered by R. S. Hall (Proc. Amer. Math. Soc. 40 (1973) 79-82). We provide an explicit calculation of the associated zeta function, which in turn allows us to obtain an expression for $N(x)$ with several terms rather than just an O-term for the error. We construct examples that illustrate a theorem of W.-B. Zhang (Illinois J. Math. 31 (1987) 645-664) about the mean value of the generalized Mobius function. We construct a continuous number system for which the error term in the integer counting function $N(x)$ oscillates as much as the de la Vallee Poussin error for the Prime Number Theorem. A simplification in a theorem of H. G. Diamond is offered. Finally we estimate the integral of a measure obtained by interpolating the integer counting measure and (essentially) the Lebesgue measure.","Made available in DSpace on 2015-09-28T15:20:17Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9737042.pdf: 2431645 bytes, checksum: 36504114a23bca171cac7c06fa3e7138 (MD5) Previous issue date: 1997","Embargo set by: Seth Robbins for item 88224 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","78 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997."],"dc:identifier":["http://hdl.handle.net/2142/86943","(MiAaPQ)AAI9737042"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["On Beurling's Theory of Generalized Numbers"],"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":"2026-07-22T22:26:28Z"}