Article
| Title: | On the least almost-prime in arithmetic progression (English) |
| Author: | Li, Jinjiang |
| Author: | Zhang, Min |
| Author: | Cai, Yingchun |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 1 |
| Year: | 2023 |
| Pages: | 177-188 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $\mathcal {P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal {P}_2(a,q)$ the least almost-prime $\mathcal {P}_2$ which satisfies $\mathcal {P}_2\equiv a\pmod q$. It is proved that for sufficiently large $q$, there holds $$ \mathcal {P}_2(a,q)\ll q^{1.8345}. $$ This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$. (English) |
| Keyword: | almost-prime |
| Keyword: | arithmetic progression |
| Keyword: | linear sieve |
| Keyword: | Selberg's $\Lambda ^2$-sieve |
| MSC: | 11N13 |
| MSC: | 11N35 |
| MSC: | 11N36 |
| idZBL: | Zbl 07655761 |
| idMR: | MR4541095 |
| DOI: | 10.21136/CMJ.2022.0478-21 |
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| Date available: | 2023-02-03T11:11:24Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151510 |
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| Reference: | [1] Halberstam, H., Richert, H.-E.: Sieve Methods.London Mathematical Society Monographs 4. Academic Press, London (1974). Zbl 0298.10026, MR 0424730 |
| Reference: | [2] Iwaniec, H.: A new form of the error term in the linear sieve.Acta Arith. 37 (1980), 307-320. Zbl 0444.10038, MR 0598883, 10.4064/aa-37-1-307-320 |
| Reference: | [3] Iwaniec, H.: On the Brun-Titchmarsh theorem.J. Math. Soc. Japan 34 (1982), 95-123. Zbl 0486.10033, MR 0639808, 10.2969/jmsj/03410095 |
| Reference: | [4] Jurkat, W. B., Richert, H.-E.: An improvement of Selberg's sieve method. I.Acta Arith. 11 (1965), 217-240. Zbl 0128.26902, MR 0202680, 10.4064/aa-11-2-217-240 |
| Reference: | [5] Levin, B. V.: On the least almost prime number in an arithmetic progression and the sequence $k^2x^2+1$.Usp. Mat. Nauk 20 (1965), 158-162 Russian. Zbl 0154.30002, MR 0188173 |
| Reference: | [6] Mertens, F.: Ein Beitrag zur analytischen Zahlentheorie: Über die Vertheilung der Primzahlen.J. Reine Angew. Math. 78 (1874), 46-63 German \99999JFM99999 06.0116.01. MR 1579612, 10.1515/crll.1874.78.46 |
| Reference: | [7] Motohashi, Y.: On almost-primes in arithmetic progressions. III.Proc. Japan Acad. 52 (1976), 116-118. Zbl 0361.10039, MR 0412128, 10.3792/pja/1195518371 |
| Reference: | [8] Pan, C. D., Pan, C. B.: Goldbach Conjecture.Science Press, Beijing (1992). Zbl 0849.11080, MR 1287852 |
| Reference: | [9] Titchmarsh, E. C.: A divisor problem.Rend. Circ. Mat. Palermo 54 (1930), 414-429 \99999JFM99999 56.0891.01. 10.1007/BF03021203 |
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