2018/11/30 by Jori Merikoski
Mathematics · #Analytic Number Theory Research #Bounded function #Constant (computer programming) #Limit (mathematics) #Limit point #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Measure (data warehouse) #Prime (order theory) #Sequence (biology) #math.NT
paper · pdf · doi:10.1112/jlms.12314
v2: small corrections, added proof of Proposition 4 in a more general case v3: Section 6 added, which contains a correction to the proofs of Lemmata 15 and 16 of the published version (this does not affect the results)
openalex created_date 2018/11/16 · openalex publication_date 2020/04/07 · arxiv created 2021/03/02 · arxiv updated 2021/03/03 · openalex updated_date 2026/08/05
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 p n denotes the nth prime and L is the set of limit points of the sequence ( p n + 1 − p n ) / log p n n = 1 ∞ , then for all T ⩾ 0 the Lebesque measure of L ∩ [ 0 , T ] is at least T / 3 . This improves the result of Pintz from 2015 that the Lebesque measure of L ∩ [ 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 ∩ [ T , T + C ] ≠ ∅ , that is, gaps between limit points are bounded by an absolute constant.