On the characterization of additive functions on residue classes (Q580394)

\textit{P. Erdős} [Ann. Math., II. Ser. 47, 1-20 (1946; Zbl 0061.079)] proved that a real-valued monotonic additive function is necessarily of the form \(c \cdot \log n.\) In an earlier note [J. Number Theory 24, 298- 304 (1986; Zbl 0597.10052)] the author generalized this result to additive functions \(f: {\mathbb{N}}\to {\mathbb{R}}^ k\) with monotonic Euclidean norm. In the paper under review the result is extended to additive functions \(f: {\mathbb{N}}\to {\mathbb{R}}^ k\) with monotonic Euclidean norm on a fixed residue class a mod q, \((a,q)=1\). In this case the author proves the existence of a constant \(c\in {\mathbb{R}}^ k\) such that \(f(n)=c \cdot \log n\) for all n which are coprime to q. The proof is similar to the one in the author's previous paper.