On the gaps between consecutive primes

: Sun Yu-Chen, Pan Hao

: 2022

Forum Mathematicum

: FORUM MATHEMATICUM

: FORUM MATH

: 14

: 0933-7741

: 1435-5337

DOI: https://doi.org/10.1515/forum-2021-0140

: https://www.degruyter.com/document/doi/10.1515/forum-2021-0140/html

: https://arxiv.org/pdf/1802.02470.pdf

Let p(n) denote the n-th prime. We prove that, for any m >= 1, there exist infinitely many n such that p(n) - p(n-m) <= C-m for some large constant C-m > 0, andp(n+1) - p(n) >= c(m) log n log log n log log log log n/log log log nfor some small constant c(m) > 0. Furthermore, for any fixed positive integer l, there are many positive integers k with (k, l) = 1 such thatp'(k, l) >= ck . log k log log k log log log log k/log log log kwhere p' (k, l) denotes the least prime of the form kn + l with n >= 1, which improves the previous result of Prachar.