Matthias Beck
Dedekind's Reciprocity Law -- The Probabilistic Way
The Dedekind sum is defined for relatively prime integers a and b as
s(a,b) = Sumk=1,...,b-1 ((ka/b)) ((k/b)),
where ((x)) = x - [x] - 1/2. This sum and its generalizations have
intrigued mathematicians from various areas such as Number Theory,
Topology, and Combinatorial Geometry since their introduction by Dedekind
in 1892. The most fundamental theorem for the Dedekind sums was already
proved by Dedekind:
s(a,b) + s(b,a) = - 1/4 + 1/12 ( a/b + 1/ab + b/a )
There are several proofs of this reciprocity law in the literature. We
will present one of the most recent ones, due to Dilcher and Girstmair,
which is based on a certain equal distribution model. We will also show
how their ideas can be generalized to a wider class of arithmetic sums.