On the characterization of additive functions on Gaussian integers (Q2714379)

Let \(G^*\) be the set of the nonzero Gaussian integers. A function \(f\) is said to be completely \(G\)-additive, if \(f(\alpha\beta)=f(\alpha)+f(\beta)\) for all \(\alpha,\beta\in G^*\). Among other results the author proves the following theorem. If \(f\) is a completely \(G\)-additive function for which \(f(\alpha^2+1)+f(\alpha^2-1)=c\) holds then \(f(\alpha)=0\) for all \(\alpha\in G^*\). Other results of this article have some common points with \textit{I. Kátai} [Characterization of log \(n\), Studies in Pure Mathematics, in Memory of P. Turán, 415-421 (1983; Zbl 0519.10044)].