An elementary proof of the Petersson-Knopp theorem on Dedekind sums
\textit{M. I. Knopp} [J. Number Theory 12, 2--9 (1980; Zbl 0423.10015)] used the functional equation of Dedekind's eta function together with Hecke operators to show that the classical Dedekind sum \(s(h,k)\) satisfies the identity \[ \sum_{_{\substack{ ad=n \\ d>0}}} \sum_{b\pmod d} s(ah+bk,dk)=\sigma(n) s(h,k) \] for arbitrary integers \(n>0\), \(h\) and \(k>0\), where \(\sigma(n)\) is the sum of the divisors of \(n\). The author gives a short elementary proof based on an identity of \textit{P. Subrahmanyam} [Math. Stud. 45, 8--12 (1977)] which expresses the inner sum as follows: \[ \sum_{b\pmod d} s(ah + bk, dk) = \sum_{c\mid d} \mu(c) s(ahc,k)\sigma(d/c). \] When this is summed over all \(d>0\) with \(ad = n\) the right member collapses to \(\sigma(n) s(h,k)\). A completely different elementary proof has been given by \textit{L. A. Parson} [Math. Proc. Camb. Philos. Soc. 88, 11--14 (1980; Zbl 0435.10005)].
scientific article; zbMATH DE number 3693353