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.