{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86952"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86952","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Analogues of Dedekind Sums","abstract":"In 1877, R. Dedekind introduced the sum$$s(d, c) = \\sum\\sbsp{j=1}{c}\\left(\\left({j\\over c}\\right)\\right)\\left(\\left({dj\\over c}\\right)\\right),$$which appears in the multiplier system of the Dedekind eta-function as a modular form. A century later, B. C. Berndt did analogous work with the classical theta-functions and introduced the sum$$S(d, c) = \\sum\\sbsp{j=1}{c-1}({-}1)\\sp{j+1+\\lbrack dj/c\\rbrack},$$among others. There is considerable literature on the Dedekind sum and its generalizations. This thesis develops the arithmetic theory of the analogous sums introduced by Berndt. Many properties, both elementary and analytic, are derived. In this thesis we parallel one of the paths laid out by H. Rademacher in his study of $s(d, c).$ New methods are required in the study of $S(d, c).$ As an application of our study of these analogous sums we give a new proof of the law of quadratic reciprocity. Finally, with a generalization of an Eisenstein series, we develop transformation formulas involving new sums that are generalizations of the Dedekind sum. We prove a reciprocity law for one of the new sums.","abstract_html":"In 1877, R. Dedekind introduced the sum$$s(d, c) = \\sum\\sbsp{j=1}{c}\\left(\\left({j\\over c}\\right)\\right)\\left(\\left({dj\\over c}\\right)\\right),$$which appears in the multiplier system of the Dedekind eta-function as a modular form. A century later, B. C. Berndt did analogous work with the classical theta-functions and introduced the sum$$S(d, c) = \\sum\\sbsp{j=1}{c-1}({-}1)\\sp{j+1+\\lbrack dj/c\\rbrack},$$among others. There is considerable literature on the Dedekind sum and its generalizations. This thesis develops the arithmetic theory of the analogous sums introduced by Berndt. Many properties, both elementary and analytic, are derived. In this thesis we parallel one of the paths laid out by H. Rademacher in his study of $s(d, c).$ New methods are required in the study of $S(d, c).$ As an application of our study of these analogous sums we give a new proof of the law of quadratic reciprocity. Finally, with a generalization of an Eisenstein series, we develop transformation formulas involving new sums that are generalizations of the Dedekind sum. We prove a reciprocity law for one of the new sums.","abstract_has_math":true,"creators":["Meyer, Jeffrey Lyle"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Berndt, B.C."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:19Z","date_published":"2015-09-28T15:20:19Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9812706"],"render_values":[{"text":"(MiAaPQ)AAI9812706","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86952","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Berndt, B.C."]},{"key":"dc:creator","label":"Author","values":["Meyer, Jeffrey Lyle"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:19Z","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/2142/86952","(MiAaPQ)AAI9812706"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In 1877, R. Dedekind introduced the sum$$s(d, c) = \\sum\\sbsp{j=1}{c}\\left(\\left({j\\over c}\\right)\\right)\\left(\\left({dj\\over c}\\right)\\right),$$which appears in the multiplier system of the Dedekind eta-function as a modular form. A century later, B. C. Berndt did analogous work with the classical theta-functions and introduced the sum$$S(d, c) = \\sum\\sbsp{j=1}{c-1}({-}1)\\sp{j+1+\\lbrack dj/c\\rbrack},$$among others. There is considerable literature on the Dedekind sum and its generalizations. This thesis develops the arithmetic theory of the analogous sums introduced by Berndt. Many properties, both elementary and analytic, are derived. In this thesis we parallel one of the paths laid out by H. Rademacher in his study of $s(d, c).$ New methods are required in the study of $S(d, c).$ As an application of our study of these analogous sums we give a new proof of the law of quadratic reciprocity. Finally, with a generalization of an Eisenstein series, we develop transformation formulas involving new sums that are generalizations of the Dedekind sum. We prove a reciprocity law for one of the new sums.","Made available in DSpace on 2015-09-28T15:20:19Z (GMT). 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Dedekind introduced the sum$$s(d, c) = \\sum\\sbsp{j=1}{c}\\left(\\left({j\\over c}\\right)\\right)\\left(\\left({dj\\over c}\\right)\\right),$$which appears in the multiplier system of the Dedekind eta-function as a modular form. A century later, B. C. Berndt did analogous work with the classical theta-functions and introduced the sum$$S(d, c) = \\sum\\sbsp{j=1}{c-1}({-}1)\\sp{j+1+\\lbrack dj/c\\rbrack},$$among others. There is considerable literature on the Dedekind sum and its generalizations. This thesis develops the arithmetic theory of the analogous sums introduced by Berndt. Many properties, both elementary and analytic, are derived. In this thesis we parallel one of the paths laid out by H. Rademacher in his study of $s(d, c).$ New methods are required in the study of $S(d, c).$ As an application of our study of these analogous sums we give a new proof of the law of quadratic reciprocity. Finally, with a generalization of an Eisenstein series, we develop transformation formulas involving new sums that are generalizations of the Dedekind sum. We prove a reciprocity law for one of the new sums.","Made available in DSpace on 2015-09-28T15:20:19Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9812706.pdf: 2786047 bytes, checksum: 41584b2499b9a3d8d60d608ac31a980c (MD5) Previous issue date: 1997","Embargo set by: Seth Robbins for item 88233 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","73 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997."],"dc:identifier":["http://hdl.handle.net/2142/86952","(MiAaPQ)AAI9812706"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Analogues of Dedekind Sums"],"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"}