On the gaps between consecutive primes

TL;DR

This paper investigates the distribution of gaps between consecutive primes, establishing bounds on small and large prime gaps, and explores related properties of least primes in arithmetic progressions.

Contribution

It proves the existence of infinitely many small prime gaps bounded by constants and provides lower bounds on large prime gaps, advancing understanding of prime distribution.

Findings

Infinitely many prime gaps are bounded by a constant C_m.

Large prime gaps grow at least as fast as a function involving logs.

Results include properties of least primes in arithmetic progressions.

Abstract

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