| Title:
|
On the least almost-prime in arithmetic progressions (English) |
| Author:
|
Wu, Liuying |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
74 |
| Issue:
|
2 |
| Year:
|
2024 |
| Pages:
|
535-548 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal P_{2}$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$, $q$ such that $(a,q)=1$, let $\mathcal P_{2}(q,a)$ denote the least $\mathcal P_{2}$ in the arithmetic progression $\{nq+a\}_{n=1}^{\infty }$. It is proved that for sufficiently large $q$, we have $$ \mathcal P_{2}(q,a)\ll q^{1.825}. $$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained $\mathcal P_{2}(q,a)\ll q^{1.8345}.$ (English) |
| Keyword:
|
almost-prime |
| Keyword:
|
arithmetic progression |
| Keyword:
|
linear sieve |
| Keyword:
|
Selberg's $\Lambda ^2$-sieve |
| MSC:
|
11N13 |
| MSC:
|
11N35 |
| MSC:
|
11N36 |
| idZBL:
|
Zbl 07893397 |
| idMR:
|
MR4764538 |
| DOI:
|
10.21136/CMJ.2024.0459-23 |
| . |
| Date available:
|
2024-07-10T14:55:42Z |
| Last updated:
|
2026-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152456 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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