I want to determine whether
is true for all sufficiently large $n\gt3$. (I don't know whether or not it's actually true, but it tests as true for a bunch of small $n$ that I've looked at).
My first thought was to use the lower-bound approximation for the $n$'th prime $n[ln(n)+ln(ln(n))-1]$ as $p_n$ on the left side, and the upper-bound approximation $n[ln(n)+ln(ln(n))]$ as $p_n$ on the right side. That at least gets rid of the primes, but of course the product on the right side still isn't smooth, so I don't know where to go from there. Also, I'm not sure if the inequality would remain true with those approximations, if it's indeed true with the primes. (I did test a bunch of small values and it seemed to remain true, but again I know that means nothing.)
Thanks for any help.