Erdős inequality for primitive sets

A set of natural numbers A is called primitive if no element of A divides any other. Let $\Omega \left(n\right)$ be the number of prime divisors of n counted with multiplicity. Let $f_z\left(A\right)={\sum }_{a\in A}\frac{z^{\Omega \left(a\right)}}{a{(\log a)}^z}$, where $z\in R_{>0}$. Erdős proved in 1935 that $f_1\left(A\right)={\sum }_{a\in A}\frac{1}{a\log a}$ is uniformly bounded over all primitive sets A. We prove a generalization of Erdős inequality which provides an analogous result for $f_z\left(A\right)$, when $z\in (0,2)$. Furthermore, we study the supremum of $f_z\left(A\right)$ over all primitive sets. We also discuss ${\lim }_{z\to 0}{f}_z\left(A\right)$, which is a generalization of Dirichlet density. We study the asymptotics of $f_z\left(P_k\right)$, where $P_k=\{n:\Omega (n)=k\}$. For $z=1$ we find the next term in asymptotic expansion of $f_1\left(P_k\right)$ refining the result of Gorodetsky, Lichtman, and Wong. We also study the supremum of ${\sum }_{a\in A}{z}^{\Omega \left(a\right)}/a$ over all primitive subsets of $[1,N]$.