2026-09-20T04:22:15Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/86952 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Berndt, B.C. Meyer, Jeffrey Lyle 2015-09-28T15:20:19Z 2015-09-28T15:20:19Z 10000-01-01 1997 1997 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). 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. http://hdl.handle.net/2142/86952 (MiAaPQ)AAI9812706 eng Mathematics Analogues of Dedekind Sums text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.