Dedekind sum

In mathematics, Dedekind sums are certain finite sums of products of a sawtooth function. Dedekind introduced them in the 1880's to express the functional equation of the Dedekind eta function, in a commentary to Bernhard Riemann's collected papers.. They have subsequently been much studied in number theory but also occur in some results in topology,
geometric combinatorics,
algebraic geometry, and
computational complexity. Dedekind sums have been generalized in various directions, satisfying a large number of functional equations; this article lists only a small fraction of these.

Definition

Define the sawtooth function as
We then define the Dedekind sum
by
For the case a = 1, one often writes

Simple formulae

Note that D is symmetric in a and b, i.e.,
and that, by the oddness of ),
By the periodicity of D in its first two arguments, the third argument being the length of the period for both,
If d is a positive integer, then
There is a proof for the last equality making use of
Furthermore, az = 1 implies D = D.

Alternative forms

If b and c are coprime, we may write s as
where the sum extends over the c-th roots of unity other than 1, i.e., over all such that and.
Equivalently, if b and c are coprime, then
This reformulation mirrors the fact that the above cotangent function is the discrete Fourier transform of the sawtooth function.

The reciprocity law

Dedekind proved that, if b and c are coprime positive integers then
There exist several proofs from first principles, and Dedekind's reciprocity law is equivalent to quadratic reciprocity.
Rewriting the reciprocity law as
it follows that the number 6c ''s is an integer.
If k'' = then
and
A relation that is prominent in the theory of the Dedekind eta function is the following. Let q = 3, 5, 7 or 13 and let n = 24/. Then given integers a, b, c, d with ad − bc = 1, with c chosen so that c = kq for some integer k > 0, define
Then is an even integer.

Rademacher's generalization of the reciprocity law

found the following generalization of the reciprocity law for Dedekind sums: If a, b, and c are pairwise coprime positive integers, then
Hence, the above triple sum vanishes if and only if is a Markov triple, i.e., a solution of the Markov equation