Erdős inequality for primitive sets

Abstract

A set of natural numbers $A$ is called primitive if no element of $A$ divides any other. Let $\hat{I}\text{c◯}(n)$ be the number of prime divisors of $n$ counted with multiplicity. Let $f_z(A)={\sum }_{a\in A}\frac{z^{\hat{I}\text{c◯}(a)}}{a(\log a{)}^z}$, where $z\in {\mathbb{R}}_{>0}$. Erdős proved in 1935 that $f_1(A)={\sum }_{a\in A}\frac{1}{a\log a}$ is uniformly bounded over all choices of primitive sets $A$. We prove the same fact for $f_z(A)$, when $z\in (0,2)$. Also we discuss the ${\lim }_{z\to 0}{f}_z(A)$. Some other results about primitive sets are generalized. In particular we study the asymptotic of $f_z({\mathbb{P}}_k)$, where ${\mathbb{P}}_k=\{n:\hat{I}\text{c◯}(n)=k\}$. In case of $z=1$ we find the next term in asymptotic expansion of $f_1({\mathbb{P}}_k)$ compared to the recent result of Gorodetsky, Lichtman, Wong.