On the gaps between consecutive primes

Abstract

Let $p_n$ denote the $n$-th prime. For any $m\ge 1$, there exist infinitely many $n$ such that $p_n-p_{n-m}\le C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\ge \frac{c_m\log n\log \log n\log \log \log \log n}{\log \log \log n},$$ for some small constant $c_m>0$. Furthermore, we also obtain a related result concerning the least primes in arithmetic progressions.

This is a very very preliminary draft, which maybe contains some mistakes