Dirichlet convolution of multiplicative functions

Question

Are there two nonzero arithmetic functions,say $f,g$, which are not multiplicative but their Dirichlet convolution is multiplicative?

Answer 1

Take an invertible non-multiplicative function $f$ and some multiplicative function $h$. Then $g = f^{-1} \ast h$ is not multiplicative (otherwise $f^{-1} = (f^{-1} \ast h) \ast h^{-1} = g \ast h^{-1}$ would be multiplicative) but $f \ast g = h$ is.