On the asymptotic density of the support of a Dirichlet convolution

Let ν be a multiplicative arithmetic function with support of positive asymptotic density. We prove that for any not identically zero arithmetic function f   such that ∑f(n)≠01/n<∞∑f(n)≠01/n<∞, the support of the Dirichlet convolution f⁎νf⁎ν 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 Möbius and Dirichlet transforms of arithmetic functions.