Files in this item

FilesDescriptionFormat

application/pdf

application/pdf9812706.pdf (3MB)Restricted to U of Illinois
(no description provided)PDF

Description

Title:Analogues of Dedekind Sums
Author(s):Meyer, Jeffrey Lyle
Doctoral Committee Chair(s):Berndt, B.C.
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
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.
Issue Date:1997
Type:Text
Language:English
Description:73 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997.
URI:http://hdl.handle.net/2142/86952
Other Identifier(s):(MiAaPQ)AAI9812706
Date Available in IDEALS:2015-09-28
Date Deposited:1997


This item appears in the following Collection(s)

Item Statistics