According to the Prime Number Theorem, a number $n$, roughly speaking, has probability of primality $\sigma_n:=1/\ln n$.
As every schoolchild learns, one can test the primality of $n$ by looking for divisibility by primes up to $\sqrt{n}$. If these divisibility tests enjoyed independence, one would get ``probability of primality" equal to $$ \tau_n:=\prod_{p\leq \sqrt{n}} \frac{p-1}{p}\ .$$
Probability interpretations aside, one can examine the ratio $\sigma_n/\tau_n$. What limit, if any, does it approach? And how fast?