On the values of Dedekind sums.

Let s ( a , b ) denote the classical Dedekind sum and S ( a , b ) = 12 s ( 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 ( q 2 − 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 ( q 2 − 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 . [ABSTRACT FROM AUTHOR]