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