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Article: An extension to the Brun-Titchmarsh theorem
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TitleAn extension to the Brun-Titchmarsh theorem
 
AuthorsChan, TH2
Choi, SKK3
Tsang, KM1
 
Issue Date2011
 
PublisherOxford University Press. The Journal's web site is located at http://qjmath.oxfordjournals.org/
 
CitationQuarterly Journal Of Mathematics, 2011, v. 62 n. 2, p. 307-322 [How to Cite?]
DOI: http://dx.doi.org/10.1093/qmath/hap045
 
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.
 
ISSN0033-5606
2012 Impact Factor: 0.557
2012 SCImago Journal Rankings: 0.442
 
DOIhttp://dx.doi.org/10.1093/qmath/hap045
 
ISI Accession Number IDWOS:000290816500003
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.

 
ReferencesReferences in Scopus
 
GrantsError Terms in the Summatory Formula for certain Arithmetical Functions
 
DC FieldValue
dc.contributor.authorChan, TH
 
dc.contributor.authorChoi, SKK
 
dc.contributor.authorTsang, KM
 
dc.date.accessioned2011-09-23T05:48:34Z
 
dc.date.available2011-09-23T05:48:34Z
 
dc.date.issued2011
 
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.
 
dc.description.naturepostprint
 
dc.identifier.citationQuarterly Journal Of Mathematics, 2011, v. 62 n. 2, p. 307-322 [How to Cite?]
DOI: http://dx.doi.org/10.1093/qmath/hap045
 
dc.identifier.citeulike9375070
 
dc.identifier.doihttp://dx.doi.org/10.1093/qmath/hap045
 
dc.identifier.eissn1464-3847
 
dc.identifier.epage322
 
dc.identifier.hkuros192206
 
dc.identifier.isiWOS:000290816500003
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.

 
dc.identifier.issn0033-5606
2012 Impact Factor: 0.557
2012 SCImago Journal Rankings: 0.442
 
dc.identifier.issue2
 
dc.identifier.scopuseid_2-s2.0-79957522067
 
dc.identifier.spage307
 
dc.identifier.urihttp://hdl.handle.net/10722/139347
 
dc.identifier.volume62
 
dc.languageeng
 
dc.publisherOxford University Press. The Journal's web site is located at http://qjmath.oxfordjournals.org/
 
dc.publisher.placeUnited Kingdom
 
dc.relation.ispartofQuarterly Journal of Mathematics
 
dc.relation.projectError Terms in the Summatory Formula for certain Arithmetical Functions
 
dc.relation.referencesReferences in Scopus
 
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License
 
dc.titleAn extension to the Brun-Titchmarsh theorem
 
dc.typeArticle
 
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<contributor.author>Choi, SKK</contributor.author>
<contributor.author>Tsang, KM</contributor.author>
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<description.abstract>The Siegel-Walfisz theorem states that for any B &gt; 0, we have &#8721;/p&#8804;x/p&#8801;a(mod k) 1 &#8764; x/&#966;(k) lox x for k &#8804; 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 &#8804; k &lt; x, &#8721;/p&#8804;x/p&#8801;a(mod k) 1 &#8810; x/&#966;(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 &#8721;/y&lt;n&#8804;x+y &#8801; a (mod k)&#969; (n) &lt; s for any x, y &gt; 0, s &#8805; 1 and 1 &#8804; k &lt; x.In particular, for s &#8804; log log (x/k), we have &#8721;/y&lt;n&#8804;x+y &#8801; a (mod k)&#969; (n) &lt; s 1 &#8810; x/&#966; (k) log (x/k) (log log (x/k) + K)s-1/(s-1)! &#8730; log log (x/k) + K and for any &#949;&#8712;(0, 1) and s &#8804; (1-&#949;) log log (x/k), we have. &#8721;/y&lt;n&#8804;x+y &#8801; a (mod k)&#969; (n) &lt; s 1 &#8810; &#949;-1x/&#966; (k) log (x/k) (log log (x/k) +K)s-1/(s-1) !. &#169; 2010. Published by Oxford University Press. All rights reserved.</description.abstract>
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Author Affiliations
  1. The University of Hong Kong
  2. University of Memphis
  3. Simon Fraser University