We consider the notions of $\nu$-separation and $(N,\nu)$-separation for increasing sequences that tend to infinity. We study several of the connections between the properties of a multiplier in the space $\mathcal{S}(\mathbb{R})$ and in other related spaces and the separation properties of the sequence of its zeros.

 
We also prove that a distributional division problem

always has tempered solutions $h$ for any tempered data $f$ if and only if the non integrable function $1/F$ admits regularizations that are tempered, and that this holds if and only if the pseudofunction $\mathcal{P}f(1/F)$ is tempered.