{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20978"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20978","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The zeta function of an order in a general algebra","abstract":"Zeta functions have been of major importance in algebraic number theory for many years. They are useful (along with L-functions) in obtaining results concerning the asymptotic distribution of ideals in a given class. In 1980 Bushnell and Reiner were able to extend these classical results to the noncommutative case by considering the zeta function of an order in a finite dimensional semisimple Q(or Q$\\sb{\\rm p}$)-algebra. What happens when the condition of semisimplicity is lifted is addressed in this thesis.","abstract_html":"Zeta functions have been of major importance in algebraic number theory for many years. They are useful (along with L-functions) in obtaining results concerning the asymptotic distribution of ideals in a given class. In 1980 Bushnell and Reiner were able to extend these classical results to the noncommutative case by considering the zeta function of an order in a finite dimensional semisimple Q(or Q$\\sb{\\rm p}$)-algebra. What happens when the condition of semisimplicity is lifted is addressed in this thesis.","abstract_has_math":true,"creators":["Seyfried, Michael David"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Janusz, Gerald"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:54:47Z","date_published":"2011-05-07T12:54:47Z","updated_at":"2026-07-22T22:25:17Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Seyfried, Michael David"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114406","(UMI)AAI9114406"],"render_values":[{"text":"AAI9114406","href":null,"code":true},{"text":"(UMI)AAI9114406","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20978","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Janusz, Gerald"]},{"key":"dc:creator","label":"Author","values":["Seyfried, Michael David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:54:47Z","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 Seyfried, Michael David"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114406","(UMI)AAI9114406","http://hdl.handle.net/2142/20978"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Zeta functions have been of major importance in algebraic number theory for many years. They are useful (along with L-functions) in obtaining results concerning the asymptotic distribution of ideals in a given class. In 1980 Bushnell and Reiner were able to extend these classical results to the noncommutative case by considering the zeta function of an order in a finite dimensional semisimple Q(or Q$\\sb{\\rm p}$)-algebra. What happens when the condition of semisimplicity is lifted is addressed in this thesis.","Two problems are discussed here: (1) How can the partial zeta function be expressed in terms of the zeta function described by Bushnell and Reiner. (2) What can be done in the case of the total zeta function. How can one deal with the infinite number of isomorphism classes that must exist here.","Analytic methods are used to answer the first question and an explicit determination of the abscissa of convergence is obtained. To answer the second question, an inductive method is developed to show how full lattices in an order can be expressed as a sum of full lattices in somewhat simpler pieces (roughly corresponding to a filtration of the Jacobson radical). This information is then used to express the total zeta function as a product of the zeta function of an order in a semisimple algebra with other factors.","Made available in DSpace on 2011-05-07T12:54:47Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114406.pdf: 1900934 bytes, checksum: a7a023c947e867c9afd66294f091f1c6 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:47:39Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:21:31-05:00 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":["The zeta function of an order in a general algebra"]}]}],"canonical_facts":{"dc:contributor":["Janusz, Gerald"],"dc:creator":["Seyfried, Michael David"],"dc:date":["2011-05-07T12:54:47Z","10000-01-01","1990"],"dc:description":["Zeta functions have been of major importance in algebraic number theory for many years. They are useful (along with L-functions) in obtaining results concerning the asymptotic distribution of ideals in a given class. In 1980 Bushnell and Reiner were able to extend these classical results to the noncommutative case by considering the zeta function of an order in a finite dimensional semisimple Q(or Q$\\sb{\\rm p}$)-algebra. What happens when the condition of semisimplicity is lifted is addressed in this thesis.","Two problems are discussed here: (1) How can the partial zeta function be expressed in terms of the zeta function described by Bushnell and Reiner. (2) What can be done in the case of the total zeta function. How can one deal with the infinite number of isomorphism classes that must exist here.","Analytic methods are used to answer the first question and an explicit determination of the abscissa of convergence is obtained. To answer the second question, an inductive method is developed to show how full lattices in an order can be expressed as a sum of full lattices in somewhat simpler pieces (roughly corresponding to a filtration of the Jacobson radical). This information is then used to express the total zeta function as a product of the zeta function of an order in a semisimple algebra with other factors.","Made available in DSpace on 2011-05-07T12:54:47Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114406.pdf: 1900934 bytes, checksum: a7a023c947e867c9afd66294f091f1c6 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:47:39Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:21:31-05:00 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":["AAI9114406","(UMI)AAI9114406","http://hdl.handle.net/2142/20978"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Seyfried, Michael David"],"dc:subject":["Mathematics"],"dc:title":["The zeta function of an order in a general algebra"],"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":"2026-07-22T22:25:17Z"}