{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/50445"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/50445","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Asymptotic formulae for certain arithmetic functions produced by fractional linear transformations","abstract":"K.T. Atanassov introduced the two arithmetic functions \\[ I(n) = \\prod_{\\nu=1}^k p_\\nu^{1/\\alpha_\\nu} \\qquad \\text{and}\\qquad R(n) = \\prod_{\\nu=1}^k p_\\nu^{\\alpha_v - 1} \\] called the irrational factor and the strong restrictive factor, respectively. A variety of authors have studied the properties of these arithmetic functions. We consider weighted combinations $I(n)^\\alpha R(n)^\\beta$ and characterize pairs $(\\alpha,\\beta)$ in order to measure how close $n$ is to being $k$-power full or $k$-power free. We then generalize these functions to a class of arithmetic functions defined in terms of fractional linear transformations arising from certain $2 \\times 2$ matrices, establish asymptotic formulae for averages of these functions, and explore certain maps that arise from considering the leading terms of these averages. We further generalize to a larger class of maps by introducing real moments, which allow us to explore new properties of these arithmetic functions. We additionally study the influence of the eigenvalues of a matrix on the associated arithmetic function, and obtain results on the local density of eigenvalues through their connection to a particular surface. Finally, we present a further generalization involving arithmetic functions defined by certain complex-valued fractional linear transformations, explore some of the properties of these new functions, and present a few open problems.","abstract_html":"K.T. Atanassov introduced the two arithmetic functions \\[ I(n) = \\prod_{\\nu=1}^k p_\\nu^{1/\\alpha_\\nu} \\qquad \\text{and}\\qquad R(n) = \\prod_{\\nu=1}^k p_\\nu^{\\alpha_v - 1} \\] called the irrational factor and the strong restrictive factor, respectively. A variety of authors have studied the properties of these arithmetic functions. We consider weighted combinations <span class=\"etd-inline-math\">I(n)<sup>\\</sup>alpha R(n)<sup>\\</sup>beta</span> and characterize pairs <span class=\"etd-inline-math\">(&alpha;,&beta;)</span> in order to measure how close $n$ is to being $k$-power full or $k$-power free. We then generalize these functions to a class of arithmetic functions defined in terms of fractional linear transformations arising from certain $2 \\times 2$ matrices, establish asymptotic formulae for averages of these functions, and explore certain maps that arise from considering the leading terms of these averages. We further generalize to a larger class of maps by introducing real moments, which allow us to explore new properties of these arithmetic functions. We additionally study the influence of the eigenvalues of a matrix on the associated arithmetic function, and obtain results on the local density of eigenvalues through their connection to a particular surface. Finally, we present a further generalization involving arithmetic functions defined by certain complex-valued fractional linear transformations, explore some of the properties of these new functions, and present a few open problems.","abstract_has_math":true,"creators":["Spiegelhalter, Paul"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Zaharescu, Alexandru","Berndt, Bruce C.","Hildebrand, A.J.","Boca, Florin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-09-16T17:17:33Z","date_published":"2014-09-16T17:17:33Z","updated_at":"2026-07-22T22:25:40Z","subjects":["Number theory","Dirichlet series","Farey fractions"],"languages":["en"],"rights":["Copyright 2014 Paul Spiegelhalter"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/50445","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zaharescu, Alexandru","Berndt, Bruce C.","Hildebrand, A.J.","Boca, Florin"]},{"key":"dc:creator","label":"Author","values":["Spiegelhalter, Paul"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-09-16T17:17:33Z","2016-09-22T20:59:27Z","2014-08","2014-09-16"]},{"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":["Number theory","Dirichlet series","Farey fractions"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Paul Spiegelhalter"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/50445"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["K.T. Atanassov introduced the two arithmetic functions \\[ I(n) = \\prod_{\\nu=1}^k p_\\nu^{1/\\alpha_\\nu} \\qquad \\text{and}\\qquad R(n) = \\prod_{\\nu=1}^k p_\\nu^{\\alpha_v - 1} \\] called the irrational factor and the strong restrictive factor, respectively. A variety of authors have studied the properties of these arithmetic functions. We consider weighted combinations $I(n)^\\alpha R(n)^\\beta$ and characterize pairs $(\\alpha,\\beta)$ in order to measure how close $n$ is to being $k$-power full or $k$-power free. We then generalize these functions to a class of arithmetic functions defined in terms of fractional linear transformations arising from certain $2 \\times 2$ matrices, establish asymptotic formulae for averages of these functions, and explore certain maps that arise from considering the leading terms of these averages. We further generalize to a larger class of maps by introducing real moments, which allow us to explore new properties of these arithmetic functions. We additionally study the influence of the eigenvalues of a matrix on the associated arithmetic function, and obtain results on the local density of eigenvalues through their connection to a particular surface. Finally, we present a further generalization involving arithmetic functions defined by certain complex-valued fractional linear transformations, explore some of the properties of these new functions, and present a few open problems.","Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-07-10T15:12:50Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Spiegelhalter_Paul.pdf: 1417493 bytes, checksum: 297993f16596438f7ff61e60ab327690 (MD5)","Made available in DSpace on 2014-09-16T17:17:33Z (GMT). No. of bitstreams: 2 Paul_Spiegelhalter.pdf: 1417483 bytes, checksum: dcbc91a7dc214b6a293b5478cf5e61ff (MD5) license.txt: 4068 bytes, checksum: 309586c51a412426a012945708741f0c (MD5)","Embargo set by: Seth Robbins for item 50556 Lift date: 2016-09-16T17:18:17Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 50556 on 2016-09-22T20:59:27Z."]},{"key":"dc:title","label":"Title","values":["Asymptotic formulae for certain arithmetic functions produced by fractional linear transformations"]}]}],"canonical_facts":{"dc:contributor":["Zaharescu, Alexandru","Berndt, Bruce C.","Hildebrand, A.J.","Boca, Florin"],"dc:creator":["Spiegelhalter, Paul"],"dc:date":["2014-09-16T17:17:33Z","2016-09-22T20:59:27Z","2014-08","2014-09-16"],"dc:description":["K.T. Atanassov introduced the two arithmetic functions \\[ I(n) = \\prod_{\\nu=1}^k p_\\nu^{1/\\alpha_\\nu} \\qquad \\text{and}\\qquad R(n) = \\prod_{\\nu=1}^k p_\\nu^{\\alpha_v - 1} \\] called the irrational factor and the strong restrictive factor, respectively. A variety of authors have studied the properties of these arithmetic functions. We consider weighted combinations $I(n)^\\alpha R(n)^\\beta$ and characterize pairs $(\\alpha,\\beta)$ in order to measure how close $n$ is to being $k$-power full or $k$-power free. We then generalize these functions to a class of arithmetic functions defined in terms of fractional linear transformations arising from certain $2 \\times 2$ matrices, establish asymptotic formulae for averages of these functions, and explore certain maps that arise from considering the leading terms of these averages. We further generalize to a larger class of maps by introducing real moments, which allow us to explore new properties of these arithmetic functions. We additionally study the influence of the eigenvalues of a matrix on the associated arithmetic function, and obtain results on the local density of eigenvalues through their connection to a particular surface. Finally, we present a further generalization involving arithmetic functions defined by certain complex-valued fractional linear transformations, explore some of the properties of these new functions, and present a few open problems.","Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-07-10T15:12:50Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Spiegelhalter_Paul.pdf: 1417493 bytes, checksum: 297993f16596438f7ff61e60ab327690 (MD5)","Made available in DSpace on 2014-09-16T17:17:33Z (GMT). No. of bitstreams: 2 Paul_Spiegelhalter.pdf: 1417483 bytes, checksum: dcbc91a7dc214b6a293b5478cf5e61ff (MD5) license.txt: 4068 bytes, checksum: 309586c51a412426a012945708741f0c (MD5)","Embargo set by: Seth Robbins for item 50556 Lift date: 2016-09-16T17:18:17Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 50556 on 2016-09-22T20:59:27Z."],"dc:identifier":["http://hdl.handle.net/2142/50445"],"dc:language":["en"],"dc:rights":["Copyright 2014 Paul Spiegelhalter"],"dc:subject":["Number theory","Dirichlet series","Farey fractions"],"dc:title":["Asymptotic formulae for certain arithmetic functions produced by fractional linear transformations"],"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":"2026-07-22T22:25:40Z"}