Number of primes between $n$ and $2n$

Question

What is a good lower bound on $\pi(2n)-\pi(n)$? Bertrand's postulate gives $1$. It is expected to be as I understand of form $\frac{c\cdot n}{\log n}$ from Prime Number Theorem.

1. Does the ratio always hold for all large enough $n$ with some $c$ always between $0$ and $c_0$ for some absolute constant $c_0$?

2. How often does it fail as far as we know?

Answer 1

This is partly answered here:

Primes between $n$ and $2n$

It follows from the prime-number theorem that $$ \lim_{n \to \infty} \frac{\pi(2n) - \pi(n)}{n/\log n} = 2 - 1 = 1,$$ so the number of primes between $n$ and $2n$, which is $\pi(2n) - \pi(n)$, is actually asymptotic to $\frac{n}{\log n}$ which gets arbitrarily large.