Jordan Ramsey

Note the following expression for $J_k(n)$ :

$J_k(n) = n^k \prod_{p|n} \left( 1-\frac1{p^k} \right)$

where the product is taken over all primes $p|n$ . Our limit is then

$\lim_{n \to \infty} \frac{J_2(n!)}{n!^2} = \lim_{n \to \infty} \prod_{p|n!} \left( 1-\frac1{p^2} \right) = \lim_{n \to \infty} \prod_{p \leq n} \left( 1-\frac1{p^2} \right) = \frac1{\zeta(2)} = \boxed{\frac6{\pi^2}}$