Bernoulli-Dedekind Sums

While studying the eta-function, Richard Dedekind derived what we today call the Dedekind Sum. The Dedekind sum is defined S(a,b)=∑_{h mod b}((h/b))((ah/b)), where a and b are positive integers and ((x))=x-Floor(x)-1/2 when x is an integer, and otherwise ((x))=0. Dedekind sums appear in many areas of mathematics, such as topology, geometric combinatorics, algorithmic complexity, algebraic geometry and modular forms, as well as exhibit many beautiful properties, the most famous being Dedekind's reciprocity law S(a,b)+S(b,a)=-1/4+(1/12)(1/a+1/b+1/(ab)) if a and b are relatively prime.

Since Dedekind, many mathematicians, like Apostol, have introduced generalizations of Dedekind sums involving Bernoulli polynomials. In 1999, a 3-variable Dedekind-like sum called the generalized Dedekind-Rademacher sum was introduced by Hall, Wilson and Zagier, as well as the reciprocity relation it satisfies. One can naturally extend the generalized Dedekind-Rademacher sum to the n-variable case and begin to ask what reciprocity law may the n-variable case satisfy.

We introduce a n-variable generalization of the generalized Dedekind-Rademacher sum we call a Bernoulli-Dedekind sum along with a corresponding reciprocity law. Our proof of the reciprocity theorm uses a novel, combinatorial approach that not only simplifies the proof of Hall, Wilson and Zagier's reciprocity theorem but also lends to the proof of an extension of Hall, Wilson and Zagier's reciprocity theorem to 4-variables.