Distributive property of completely multiplicative functions (Q619338)

It is well known that an arithmetic function \(f\) is completely multiplicative if and only if \(f(g\ast h)\) for all arithmetic functions \(g\) and \(h\), where \(\ast\) is the Dirichlet convolution. The authors consider this property with \(h=\mu_{\alpha}\), the Souriau-Hsu-Möbius function, and they apply the concepts of discriminative and partially discriminative products by \textit{E. Langford} [Am. Math. Mon. 80, 411--414 (1973; Zbl 0267.10004)]. The authors also show that the product \(\Lambda_{k}^{(\alpha)}\ast\mu_{-\alpha}\) is semi-discriminative, where \(\Lambda_{k}^{(\alpha)}\) is a generalized von Mangoldt function. The concept of a semi-discriminative product is given in [the reviewer, Number theory. Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri, Turku, Finland, 1999. Berlin: de Gruyter, 115--123 (2001; Zbl 0972.11006)]. The authors apply this property to derive a generalization of the Ivić-Haukkanen characterization of completely multiplicative functions [\textit{A. Ivić}, Math. Balk. 3, 158--165 (1973; Zbl 0284.10018)].