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.
Define the sawtooth function as
We then let
- D :Z3 → R
be defined by
the terms on the right being the Dedekind sums. For the case a=1, one often writes
- s(b,c) = D(1,b;c).
Note that D is symmetric in a and b, and hence
and that, by the oddness of (()),
- D(−a,b;c) = −D(a,b;c),
- D(a,b;−c) = D(a,b;c).
By the periodicity of D in its first two arguments, the third argument being the length of the period for both,
- D(a,b;c)=D(a+kc,b+lc;c), for all integers k,l.
If d is a positive integer, then
- D(ad,bd;cd) = dD(a,b;c),
- D(ad,bd;c) = D(a,b;c), if (d,c) = 1,
- D(ad,b;cd) = D(a,b;c), if (d,b) = 1.
There is a proof for the last equality making use of
Furthermore, az = 1 (mod c) implies D(a,b;c) = D(1,bz;c).
If b and c are coprime, we may write s(b,c) 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
If b and c are coprime positive integers then
Rewriting this as
it follows that the number 6c s(b,c) is an integer.
If k = (3, c) then
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/(q − 1). Then given integers a, b, c, d with ad − bc = 1 (thus belonging to the modular group), 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
- Rademacher, Hans (1954). "Generalization of the reciprocity formula for Dedekind sums". Duke Mathematical Journal. 21: 391–397. doi:10.1215/s0012-7094-54-02140-7. Zbl 0057.03801.
- Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 (See chapter 3.)
- Matthias Beck and Sinai Robins, Dedekind sums: a discrete geometric viewpoint, (2005 or earlier)
- Hans Rademacher and Emil Grosswald, Dedekind Sums, Carus Math. Monographs, 1972. ISBN 0-88385-016-8.