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LIMIT POINTS AND LONG GAPS BETWEEN PRIMES (2016)

Baker, R., Freiberg, T.

Oxford University Press

In: Quarterly Journal of Mathematics

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Publication Date: 2016-05-31

Description: Let $d_n := p_{n+1} - p_n$ , where $p_n$ denotes the $n$ th smallest prime, and let $R(T) := \log T \log _2 T\log _4 T/ (\log _3 T)^2$ (the ‘Erdös–Rankin’ function). We consider the sequence $(d_n/R(p_n))$ of normalized prime gaps, and show that its limit point set contains at least $25\%$ of non-negative real numbers. We also show that the same result holds if $R(T)$ is replaced by any ‘reasonable’ function that tends to infinity more slowly than $R(T)\log _3 T$ . We also consider ‘chains’ of normalized prime gaps. Our proof combines breakthrough work of Maynard and Tao on bounded gaps between primes with subsequent developments of Ford, Green, Konyagin, Maynard and Tao on long gaps between consecutive primes.

Print ISSN: 0033-5606

Electronic ISSN: 1464-3847

Topics: Mathematics

Published by Oxford University Press

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