A1 Refereed original research article in a scientific journal
Limit points of normalized prime gaps
Authors: Jori Merikoski
Publisher: WILEY
Publication year: 2020
Journal: Journal of the London Mathematical Society
Journal name in source: JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
Journal acronym: J LOND MATH SOC
Volume: 102
Issue: 1
Number of pages: 26
ISSN: 0024-6107
eISSN: 1469-7750
DOI: https://doi.org/10.1112/jlms.12314(external)
Self-archived copy’s web address: http://arxiv.org/pdf/1811.03008(external)
Abstract
We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if pn denotes the nth prime and L is the set of limit points of the sequence {(pn+1-pn)/logpn}n=1 infinity, then for all T > 0 the Lebesque measure of L boolean AND[0,T] is at least T/3. This improves the result of Pintz from 2015 that the Lebesque measure of L boolean AND[0,T] is at least (1/4-o(1))T, which was obtained by a refinement of the previous ideas of Banks, Freiberg, and Maynard from 2015. Our improvement comes from using Chen's sieve to give, for a certain sum over prime pairs, a better upper bound than what can be obtained using Selberg's sieve. Even though this improvement is small, a modification of the arguments given by Pintz and Banks, Freiberg, and Maynard shows that this is sufficient. In addition, we show that there exists a constant C such that for all T > 0 we have L boolean AND[T,T+C]not equal null , that is, gaps between limit points are bounded by an absolute constant.
We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if pn denotes the nth prime and L is the set of limit points of the sequence {(pn+1-pn)/logpn}n=1 infinity, then for all T > 0 the Lebesque measure of L boolean AND[0,T] is at least T/3. This improves the result of Pintz from 2015 that the Lebesque measure of L boolean AND[0,T] is at least (1/4-o(1))T, which was obtained by a refinement of the previous ideas of Banks, Freiberg, and Maynard from 2015. Our improvement comes from using Chen's sieve to give, for a certain sum over prime pairs, a better upper bound than what can be obtained using Selberg's sieve. Even though this improvement is small, a modification of the arguments given by Pintz and Banks, Freiberg, and Maynard shows that this is sufficient. In addition, we show that there exists a constant C such that for all T > 0 we have L boolean AND[T,T+C]not equal null , that is, gaps between limit points are bounded by an absolute constant.
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