| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2023-02-03T11:11:24Z |
| Last updated:
|
2025-04-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151510 |
| . |
| 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 |
| . |
