Define the sawtooth function as
- ;\\0,&{\mbox{if }}x\in \mathbb {Z} .\end{cases}}}
We then let
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
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/(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 nδ is an even integer.
Hans Rademacher found the following generalization of the reciprocity law for Dedekind sums:[1] If a, b, and c are pairwise coprime positive integers, then
Hence, the above triple sum vanishes if and only if (a, b, c) is a Markov triple, i.e. a solution of the Markov equation