Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdős, we show that \[ G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},\] where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.
Kevin Ford 1 ; Ben Green 2 ; Sergei Konyagin 3 ; Terence Tao 4
@article{10_4007_annals_2016_183_3_4,
author = {Kevin Ford and Ben Green and Sergei Konyagin and Terence Tao},
title = {Large gaps between consecutive prime numbers},
journal = {Annals of mathematics},
pages = {935--974},
year = {2016},
volume = {183},
number = {3},
doi = {10.4007/annals.2016.183.3.4},
mrnumber = {3488740},
zbl = {1338.11083},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2016.183.3.4/}
}
TY - JOUR AU - Kevin Ford AU - Ben Green AU - Sergei Konyagin AU - Terence Tao TI - Large gaps between consecutive prime numbers JO - Annals of mathematics PY - 2016 SP - 935 EP - 974 VL - 183 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2016.183.3.4/ DO - 10.4007/annals.2016.183.3.4 LA - en ID - 10_4007_annals_2016_183_3_4 ER -
%0 Journal Article %A Kevin Ford %A Ben Green %A Sergei Konyagin %A Terence Tao %T Large gaps between consecutive prime numbers %J Annals of mathematics %D 2016 %P 935-974 %V 183 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2016.183.3.4/ %R 10.4007/annals.2016.183.3.4 %G en %F 10_4007_annals_2016_183_3_4
Kevin Ford; Ben Green; Sergei Konyagin; Terence Tao. Large gaps between consecutive prime numbers. Annals of mathematics, Tome 183 (2016) no. 3, pp. 935-974. doi: 10.4007/annals.2016.183.3.4
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