If $p_n$ denotes the $n$-th prime number, we can define $$g_n:=p_{n+1}-p_n$$ as the gap after the $n$-th prime number. The merit of a prime gap is defined as $$m(p_n):=\frac{g_n}{\ln(p_n)}$$
It is well known that the merit of a prime gap can become arbitary large.
How can this however be proven ?
I do not think, it can be derived from the prime-number theorem. What else can we do ?