Chains of large gaps between primes

Kevin Ford, James Maynard, Terence Tao

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Let p n denote the n-th prime, and for any k ≥ 1 and sufficiently large X, define the quantity Gk(X) := max pn+k≤X min(pn+1 - p n , p n+k - pn+k-1), which measures the occurrence of chains of k consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that G 1 (X) » logX log log X log log log logX/log log log X for sufficiently large X. In this note, we combine the arguments in that paper with the Maier matrix method to show that G k (X) » 1/k 2 logX log log X log log log log X/log log logX for any fixed k and sufficiently large X. The implied constant is effective and independent of k.

Original languageEnglish (US)
Title of host publicationIrregularities in the Distribution of Prime Numbers
Subtitle of host publicationFrom the Era of Helmut Maier's Matrix Method and Beyond
PublisherSpringer
Pages1-21
Number of pages21
ISBN (Electronic)9783319927770
ISBN (Print)9783319927763
DOIs
StatePublished - Jul 4 2018

ASJC Scopus subject areas

  • General Mathematics

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