On the asymptotic density of the support of a Dirichlet convolution

Let $\nu$ be a multiplicative arithmetic function with support of positive asymptotic density. We prove that for any not identically zero arithmetic function $f$ such that $\sum_{f(n) \neq 0} 1 / n < \infty$, the support of the Dirichlet convolution $f * \nu$ possesses a positive asymptotic density. When $f$ is a multiplicative function, we give also a quantitative version of this claim. This generalizes a previous result of P. Pollack and the author, concerning the support of M\"obius and Dirichlet transforms of arithmetic functions.