Dedekind sum


In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have subsequently been much studied in number theory, and have occurred in some problems of topology. Dedekind sums have a large number functional equations; this article lists only a small fraction of these.
Dedekind sums were introduced by Richard Dedekind in a commentary on fragment XXVIII of Bernhard Riemann's collected papers.

Definition

Define the sawtooth function as
We then let
be defined by
the terms on the right being the Dedekind sums. For the case a=1, one often writes

Simple formulae

Note that D is symmetric in a and b, and hence
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.
If b, c > 0 are coprime, then

Reciprocity law

If b and c are coprime positive integers then
Rewriting this 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 one has nδ 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