Extension of the Liouville and Skoruppa arithmetic identities (Q610595)

Let \(\Omega\) be the set of all \(2\times 2\) matrices with integer entries, \(U:\Omega\to \Omega\), \[ U\left(\begin{pmatrix} a&a'\\b&b'\end{pmatrix}\right)=\begin{pmatrix} a'&a'-a\\-b'&b'+b\end{pmatrix}, \] \(A\) an abelian group, \(\Phi:\Omega\to A\) a function such that \(\Phi(U\omega)=\Phi(\omega)\) for any \(\omega\in\Omega\). The author proves that for any positive integer \(d\) \[ \begin{multlined} \sum_{mn'+m'n=d}\left(\Phi\left(\begin{pmatrix} m&m'\\n&n'\end{pmatrix}\right)-\Phi\left(\begin{pmatrix} m&-m'\\-n&n'\end{pmatrix}\right)\right)\\ =\sum_{d_1d_2=d}\left(\sum_{0\leq a<d_2}\Phi\left(\begin{pmatrix} 0&d_1\\d_2&-a\end{pmatrix}\right)-\sum_{0\leq b<d_1}\Phi\left(\begin{pmatrix} b&d_1\\d_2&0\end{pmatrix}\right)\right).\end{multlined} \] As particular cases one obtains some arithmetic identities of Liouville and Skoruppa.