A lower bound for the least prime in an arithmetic progression

@article{Li2016ALB,
  title={A lower bound for the least prime in an arithmetic progression},
  author={Junxian Li and Kyle Pratt and George Shakan},
  journal={arXiv: Number Theory},
  year={2016}
}
Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular, we show that for almost every $k$ one has $P(k) \gg \phi(k) \log k \log_2 k \log_4 k / \log_3 k,$ answering a question of Ford, Green, Konyangin, Maynard, and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve… Expand

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