Limit of Difference between Consecutive Prime Numbers

Theorem
The Prime Number Theorem indicates that the average value of the difference between two consecutive prime numbers is of the order of $\log p_n$.

Let $E = \displaystyle \liminf_{n \mathop \to \infty} \dfrac {p_{n + 1} - p_n} {\log p_n}$.

If there are infinitely many twin primes, then $E = 0$.

If not, then it is not known what the value of $E$ is.