On the values of the Dedekind sum
Let \(S(h,k)\) be the Dedekind sum. We define \(t(h,k)=6k\, s(h,k)\). It is known that \(t(h,k)\) is an integer for all \(h\) and \(k\). The problem we address is that of characterizing, for a given integer \(t\), all pairs \((h,k)\) such that \(t(h,k)=t\). Assuming, without significant loss of generality, that \(t\geq 0\) and that \(h\) and \(k\) are relatively prime, we prove: Theorem. Given an integer \(t\), there exists a finite, computable (and possibly empty) set \(Q_ t\) of binary quadratic forms of discriminant \(4t^ 2-4\) such that \(t(h,k)=t\) if and only if there exist \(Q\) in \(Q_ t\) and \(u,v,x,y\) in \({\mathbb Z}\) with \(uy-vx=1\), \(k=Q(x,y)\), and \(h=t-B(u,v;x,y)\); here, \(B\) is the symmetric bilinear form such that \(B(x,y;x,y)=Q(x,y).\) As an application, we show that \(t(h,k)=3\) if and only if there exist integers \(u,v,x\) and \(y\) with \(uy-vx=1\) such that \(h=ux-8vy+3\) and \(k=8y^ 2-x^ 2>0\). This paper can be seen as an extension of results of \textit{H. Salié} [Math. Z. 72, 61--75 (1959; Zbl 0085.26804).
scientific article; zbMATH DE number 3867417