Article
| 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 |
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| Category: | math |
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| 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 |
| DOI: | 10.21136/CMJ.2024.0459-23 |
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| Date available: | 2024-07-10T14:55:42Z |
| Last updated: | 2024-07-15 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152456 |
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