The number of multiplicative Sidon sets of integers

A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erd\H{o}s in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the lower order term: $\pi(n)+\Theta(\frac{n^{3/4}}{(\log n)^{3/2}})$. We show that the number of multiplicative Sidon subsets of $\{1,\ldots, n\}$ is $T(n)\cdot 2^{\Theta(\frac{n^{3/4}}{(\log n)^{3/2}})}$ for a certain function $T(n)\approx 2^{1.815\pi(n)}$ which we specify. This is a rare example in which the order of magnitude of the lower order term in the exponent is determined. It resolves the enumeration problem for multiplicative Sidon sets initiated by Cameron and Erd\H{o}s in the 80s. We also investigate its extension for generalised multiplicative Sidon sets. Denote by $S_k$, $k\ge 2$, the number of multiplicative $k$-Sidon subsets of $\{1,\ldots, n\}$. We show that $S_k(n)=(\beta_k+o(1))^{\pi(n)}$ for some $\beta_k$ we define explicitly. Our proof is elementary.