Remark on multiplicative functions (Q2554350)

In this note it is proved that if \(f\) is a multiplicative function and for some \(k\) we have \(f(k)\ne 0\), then \[ F_k(n) = f(kn)/f(k) \text{ is multiplicative}.\] For \(f = \varphi\) (Euler's totient) this result was obtained by \textit{D. H. Lehmer} and \textit{E. Lehmer} [Stud. Math. Anal. related Topics, Essays in Honor of G. Polýa, 202--210 (1962; Zbl 0129.02706)].