Article: An extension to the Brun-Titchmarsh theorem
| Title | An extension to the Brun-Titchmarsh theorem | ||||||
|---|---|---|---|---|---|---|---|
| Authors | Chan, TH2 Choi, SKK3 Tsang, KM1 | ||||||
| Issue Date | 2011 | ||||||
| Publisher | Oxford 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?] DOI: http://dx.doi.org/10.1093/qmath/hap045 | ||||||
| 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<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. | ||||||
| ISSN | 0033-5606 2011 Impact Factor: 0.617 2011 SCImago Journal Rankings: 0.033 | ||||||
| DOI | http://dx.doi.org/10.1093/qmath/hap045 | ||||||
| ISI Accession Number ID | WOS:000290816500003
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 | References in Scopus | ||||||
| Grants | Error Terms in the Summatory Formula for certain Arithmetical Functions |
| dc.contributor.author | Chan, TH | ||||||
|---|---|---|---|---|---|---|---|
| dc.contributor.author | Choi, SKK | ||||||
| dc.contributor.author | Tsang, KM | ||||||
| dc.date.accessioned | 2011-09-23T05:48:34Z | ||||||
| dc.date.available | 2011-09-23T05:48:34Z | ||||||
| dc.date.issued | 2011 | ||||||
| dc.description.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<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.grant | Error Terms in the Summatory Formula for certain Arithmetical Functions | ||||||
| dc.description.grantcode | 9940 | ||||||
| dc.description.nature | Link_to_subscribed_fulltext | ||||||
| dc.identifier.citation | Quarterly 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.citeulike | 9375070 | ||||||
| dc.identifier.doi | http://dx.doi.org/10.1093/qmath/hap045 | ||||||
| dc.identifier.epage | 322 | ||||||
| dc.identifier.hkuros | 192206 | ||||||
| dc.identifier.isi | WOS:000290816500003
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.issn | 0033-5606 2011 Impact Factor: 0.617 2011 SCImago Journal Rankings: 0.033 | ||||||
| dc.identifier.issue | 2 | ||||||
| dc.identifier.scopus | eid_2-s2.0-79957522067 | ||||||
| dc.identifier.spage | 307 | ||||||
| dc.identifier.uri | http://hdl.handle.net/10722/139347 | ||||||
| dc.identifier.volume | 62 | ||||||
| dc.language | eng | ||||||
| dc.publisher | Oxford University Press. The Journal's web site is located at http://qjmath.oxfordjournals.org/ | ||||||
| dc.publisher.place | United Kingdom | ||||||
| dc.relation.ispartof | Quarterly Journal of Mathematics | ||||||
| dc.relation.references | References in Scopus | ||||||
| dc.title | An extension to the Brun-Titchmarsh theorem | ||||||
| dc.type | Article |
Author Affiliations
- The University of Hong Kong
- University of Memphis
- Simon Fraser University