Whats the most amount of primes we can prove the existence of, that are less than some prime number squared?

Question

Given some arbitrary prime number $p_n$

How many prime numbers can we prove exist which are smaller than ${p_n}^2?$

...Obviously we can say that $n$ primes must exist, but I think I can prove that $n + p_n - 1$ prime numbers must exists which are smaller than ${p_n}^2$.

Are there any better results?

Answer 1

The prime number theorem gives much better bounds. Here is a screenshot of some of those bounds, for those who can't use the link. The famous paper of Rosser and Schoenfeld is a good source for such inequalities.