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Article: An extension to the Brun-Titchmarsh theorem

TitleAn extension to the Brun-Titchmarsh theorem
Authors
Issue Date2011
PublisherOxford University Press. The Journal's web site is located at http://qjmath.oxfordjournals.org/
Citation
Quarterly Journal Of Mathematics, 2011, v. 62 n. 2, p. 307-322 How to Cite?
Abstract
The Siegel-Walfisz theorem states that for any B > 0, we have ∑/p≤x/p≡a(mod k) 1 ∼ x/φ(k) lox x for k ≤ log B x and (k, a) = 1. This only gives an asymptotic formula for the number of primes over an arithmetic progression for quite small moduli k compared with x. However, if we are only concerned about upper bound, we have the Brun-Titchmarsh theorem, namely for any 1 ≤ k < x, ∑/p≤x/p≡a(mod k) 1 ≪ x/φ(k) lox (x/k) In this article, we prove an extension to the Brun-Titchmarsh theorem on the number of integers, with at most s prime factors, in an arithmetic progression, namely ∑/y 0, s ≥ 1 and 1 ≤ k < x.In particular, for s ≤ log log (x/k), we have ∑/y
Persistent Identifierhttp://hdl.handle.net/10722/139347
ISSN
2013 Impact Factor: 0.593
ISI Accession Number ID
Funding AgencyGrant Number
NSERC of Canada
RGC of Hong Kong, SAR, ChinaHKU 7042/04P
Funding Information:

Research of K.K.C. was supported by NSERC of Canada. Research of K.M.T. was fully supported by RGC grant HKU 7042/04P of Hong Kong, SAR, China.

References
Grants

 

DC FieldValueLanguage
dc.contributor.authorChan, THen_HK
dc.contributor.authorChoi, SKKen_HK
dc.contributor.authorTsang, KMen_HK
dc.date.accessioned2011-09-23T05:48:34Z-
dc.date.available2011-09-23T05:48:34Z-
dc.date.issued2011en_HK
dc.identifier.citationQuarterly Journal Of Mathematics, 2011, v. 62 n. 2, p. 307-322en_HK
dc.identifier.issn0033-5606en_HK
dc.identifier.urihttp://hdl.handle.net/10722/139347-
dc.description.abstractThe Siegel-Walfisz theorem states that for any B > 0, we have ∑/p≤x/p≡a(mod k) 1 ∼ x/φ(k) lox x for k ≤ log B x and (k, a) = 1. This only gives an asymptotic formula for the number of primes over an arithmetic progression for quite small moduli k compared with x. However, if we are only concerned about upper bound, we have the Brun-Titchmarsh theorem, namely for any 1 ≤ k < x, ∑/p≤x/p≡a(mod k) 1 ≪ x/φ(k) lox (x/k) In this article, we prove an extension to the Brun-Titchmarsh theorem on the number of integers, with at most s prime factors, in an arithmetic progression, namely ∑/y<n≤x+y ≡ a (mod k)ω (n) < s for any x, y > 0, s ≥ 1 and 1 ≤ k < x.In particular, for s ≤ log log (x/k), we have ∑/y<n≤x+y ≡ a (mod k)ω (n) < s 1 ≪ x/φ (k) log (x/k) (log log (x/k) + K)s-1/(s-1)! √ log log (x/k) + K and for any ε∈(0, 1) and s ≤ (1-ε) log log (x/k), we have. ∑/y<n≤x+y ≡ a (mod k)ω (n) < s 1 ≪ ε-1x/φ (k) log (x/k) (log log (x/k) +K)s-1/(s-1) !. © 2010. Published by Oxford University Press. All rights reserved.en_HK
dc.languageengen_US
dc.publisherOxford University Press. The Journal's web site is located at http://qjmath.oxfordjournals.org/en_HK
dc.relation.ispartofQuarterly Journal of Mathematicsen_HK
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.titleAn extension to the Brun-Titchmarsh theoremen_HK
dc.typeArticleen_HK
dc.identifier.emailTsang, KM:kmtsang@maths.hku.hken_HK
dc.identifier.authorityTsang, KM=rp00793en_HK
dc.description.naturepostprint-
dc.identifier.doi10.1093/qmath/hap045en_HK
dc.identifier.scopuseid_2-s2.0-79957522067en_HK
dc.identifier.hkuros192206en_US
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-79957522067&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume62en_HK
dc.identifier.issue2en_HK
dc.identifier.spage307en_HK
dc.identifier.epage322en_HK
dc.identifier.eissn1464-3847-
dc.identifier.isiWOS:000290816500003-
dc.publisher.placeUnited Kingdomen_HK
dc.relation.projectError Terms in the Summatory Formula for certain Arithmetical Functions-
dc.identifier.scopusauthoridChan, TH=7402680875en_HK
dc.identifier.scopusauthoridChoi, SKK=7408121473en_HK
dc.identifier.scopusauthoridTsang, KM=7201554731en_HK
dc.identifier.citeulike9375070-

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