Large gaps between primes

@article{Maynard2014LargeGB,
  title={Large gaps between primes},
  author={James Maynard},
  journal={arXiv: Number Theory},
  year={2014}
}
  • J. Maynard
  • Published 1 August 2014
  • Mathematics
  • arXiv: Number Theory
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos. 
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Large gaps between consecutive prime numbers
Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log
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