Small gaps between products of two primes

Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 − qn ≤ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if ν is any positive integer, then (qn+ν − qn) ≤ ν eν − γ (1+o(1)) in...

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Vydáno v:	Proceedings of the London Mathematical Society Ročník 98; číslo 3; s. 741 - 774
Hlavní autoři:	Goldston, D. A., Graham, S. W., Pintz, J., Y ld r m, C. Y.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Oxford University Press 01.05.2009
ISSN:	0024-6115, 1460-244X
On-line přístup:	Získat plný text
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Popis
Shrnutí:	Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 − qn ≤ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if ν is any positive integer, then (qn+ν − qn) ≤ ν eν − γ (1+o(1)) infinitely often. We also prove several other related results on the representation of numbers with exactly two prime factors by linear forms.
Bibliografie:	ark:/67375/HXZ-K24WJJM0-X ArticleID:pdn046 2000 Mathematics Subject Classification 11N25 (primary), 11N36 (secondary). istex:5D881B3214CA89DA97F8495611F149FDEE44B8CD
ISSN:	0024-6115 1460-244X
DOI:	10.1112/plms/pdn046