How one can get an estimate or a range for the integer $n$

Question

Let $(p_{n})_{n≥1}$ be the sequence of prime numbers. I have an inequality of the form: $$p_{n}<a$$ where $a$ is a real number.

My question is: How one can get an estimate or a range for the integer $n$.

Answer 1

If I understand correctly, you are asking about approximating the prime counting function, which gives the number of prime numbers such that $p\leq a$. There exist many approximations for the prime counting function, including $n\approx\frac{a}{\ln a}$