I came up with the following question, and I don't have proof of it.
Let $m>1$ be a positive integer. Let $f:\mathbb{N}\to \mathbb{C}$ a multiplicative function, whose image is a subset of the $m$-th roots of unity. Then the limit $$\lim_{N\to +\infty}\frac{\sum_{k=1}^Nf(k)}{N}$$ exists.