Let $f:\mathbb N\to \mathbb C$ be some arithmetical function. Define $\varphi_f(n)$ by the following formula:
$$ \varphi_f(n)=\sum_{\substack{k\leq n \\ (k,n)=1}}f(k). $$
In other words, $\varphi_f(n)$ is the sum of $f$ over totatives of $n$. For example, if $f=\delta_1(n)$ then $\varphi_f(n)=1$, if $f=1$ then $\varphi_f(n)=\varphi(n) -$ Euler's totient function. Assume that $f$ is completely multiplicative. Examination of the first few values of $f$ (up to $\approx 40$) shows that if $\varphi_f$ is multiplicative only if $f=1$ or $f=\delta_1$.
Obviously, direct analysis of 40 or so cases is not the most illuminating way to prove this type of proposition. This leads us to a bit more general question. Let us call an arithmetical function $g$ eventually multiplicative if there is a multiplicative function $G$ such that $g(n)=G(n)$ for $n$ large enough. Is it true that if $f$ is completely multiplicative and $\varphi_f$ is eventually multiplicative then either $f=\delta_1$ or $f=1$?
