Erd{o}s inequality for primitive sets

A set of natural numbers A is called primitive if no element of A divides any other. Let Ω(n) be the number of prime divisors of n counted with multiplicity. Let fz(A) = ∑a ∈ AzΩ(a)a (log a)ᶻ, where z ∈ R> 0. Erd{o}s proved in 1935 that f₁(A) = ∑a ∈ A1/a log a is uniformly bounded over all choices of primitive sets A. We prove the same fact for fz(A), when z ∈ (0, 2). Also we discuss the limz → 0 fz(A). Some other results about primitive sets are generalized. In particular we study the asymptotic of fz(Pₖ), where Pₖ = \ n : Ω(n) = k \. In case of $z = 1$ we find the next term in asymptotic expansion of f₁(Pₖ) compared to the recent result of Gorodetsky, Lichtman, Wong.