On the values of Dedekind sums

Let s(a,b) denote the classical Dedekind sum and S(a,b)=12s(a,b). For a given denominator q∈N, we study the numerators k∈Z of the values k/q, (k,q)=1, of Dedekind sums S(a,b). Our main result says that if k is such a numerator, then the whole residue class of k modulo (q2−1)q consists of numerators of this kind. This fact reduces the task of finding all possible numerators k to that of finding representatives for finitely many residue classes modulo (q2−1)q. By means of the proof of this result we have determined all possible numerators k for 2≤q≤60, the case q=1 being trivial. The result of this search suggests a conjecture about all possible values k/q, (k,q)=1, of Dedekind sums S(a,b) for an arbitrary q∈N.