Let $n$ be a positive integer and let $p(n)$ be the $n$th prime. Let $$f(n) = \dfrac{1}{30} \prod_{3<i<n+1} \left(\dfrac{p(i)- \left( \dfrac{2i}{\ln(p(i))}\right) + 1}{p(i)} \right).$$
How does $f(n)$ behave asymptotically? Does $\lim_{n\to oo} (n+7)^2 f(n)$ exist and what value is it? Can the limit be given in closed form?