On infinite multiplicative Sidon sets

We prove that if $A$ is an infinite multiplicative Sidon set, then $\liminf\limits_{n\to \infty}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n)^3}}<\infty$andconstructaninfinitemultiplicativeSidonsetsatisfying$\liminf\limits_{n\to \infty}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n)^3}}>0$.