On a class of \(\psi\)-products preserving multiplicativity
\textit{D. H. Lehmer} [Trans. Am. Math. Soc. 33, 945-957 (1931; Zbl 0003.10201)] defined the \(\psi\)-product of two arithmetic functions \(f\) and \(g\) by \((f\psi g)(n)=\sum_{\psi(x,y)=n}f(x)g(y)\) where \(\psi\) is a mapping defined on a nonempty subset \(T\) of \(\mathbb{N}\times\mathbb{N}\) and satisfying for example the conditions: \(\psi(x,y)\geq\max(x,y)\) and \(\psi(1,k)=k\) for each \(k\in\mathbb{N}\). A special case of \(\psi\)-products is given by Dirichlet's convolution \(f*g\) obtained by setting \(\psi(x,y)=x\cdot y\). It is well known that \(f*g\) is multiplicative whenever \(f\) and \(g\) are. In the present paper the authors characterize all multiplicativity preserving \(\psi\)- functions.
scientific article; zbMATH DE number 35121