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arXiv · 1707.05437

Bounded gaps between primes in short intervals

Ryan Alweiss·Sammy Luo
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Abstract

Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any $\delta\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^\delta,x]$ contains $\gg_{k} \frac{x^\delta}{(\log x)^k}$ pairs of consecutive primes differing by at most $d$. This confirms a speculation of Maynard that results on small gaps between primes can be refined to the setting of short intervals of this length.

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Ryan Alweiss, Sammy Luo. 2017-07-18. Bounded gaps between primes in short intervals. https://doi.org/10.1007/s40993-018-0109-y

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