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 language | English (US) |
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Title of host publication | Irregularities in the Distribution of Prime Numbers |
Subtitle of host publication | From the Era of Helmut Maier's Matrix Method and Beyond |
Publisher | Springer |
Pages | 1-21 |
Number of pages | 21 |
ISBN (Electronic) | 9783319927770 |
ISBN (Print) | 9783319927763 |
DOIs | |
State | Published - Jul 4 2018 |
ASJC Scopus subject areas
- General Mathematics