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Long gaps in sieved sets

Abstract:

For each prime p, let Ip⊂Z/pZ denote a collection of residue classes modulo p such that the cardinalities |Ip| are bounded and about 1 on average. We show that for sufficiently large x, the sifted set {n∈Z:n(modp)∉Ipforallp≤x} contains gaps of size x(logx)δ depends only on the densitiy of primes for which Ip≠∅. This improves on the "trivial'' bound of ≫x. As a consequence, for any non-constant polynomial f:Z→Z with positive leading coefficient, the set {n≤X:f(n)composite} contains an interval...

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Publication status:
Published
Peer review status:
Peer reviewed

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Publisher copy:
10.4171/JEMS/1020

Authors


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Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Oxford college:
Magdalen College
Role:
Author
ORCID:
0000-0001-5782-7082
Publisher:
European Mathematical Society Publisher's website
Journal:
Journal of the European Mathematical Society Journal website
Volume:
23
Issue:
2
Pages:
667-700
Publication date:
2020-11-15
Acceptance date:
2020-01-19
DOI:
EISSN:
1435-9863
ISSN:
1435-9855
Language:
English
Keywords:
Pubs id:
pubs:1083212
UUID:
uuid:07acb41c-02b4-4ab9-bd7b-eafb315e1c8e
Local pid:
pubs:1083212
Source identifiers:
1083212
Deposit date:
2020-01-19

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