Analogues of Dedekind Sums

Meyer, Jeffrey Lyle

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Description

Title

Analogues of Dedekind Sums

Author(s)

Meyer, Jeffrey Lyle

Issue Date

1997

Doctoral Committee Chair(s)

Berndt, B.C.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

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.

Graduation Semester

1997

Type of Resource

text

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