They have subsequently been much studied in number theory, and have occurred in some problems of topology.
Dedekind sums have a large number of functional equations; this article lists only a small fraction of these.
For the case a = 1, one often writes 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 (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
If k = (3, c) then and A relation that is prominent in the theory of the Dedekind eta function is the following.
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