Geodesic
On the least almost-prime in arithmetic progressions
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 535-548 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\mathcal P_{2}(q,a)\ll q^{1.8345}.$
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 $\mathcal P_{2}(q,a)\ll q^{1.8345}.$
DOI : 10.21136/CMJ.2024.0459-23
Classification : 11N13, 11N35, 11N36
Keywords: almost-prime; arithmetic progression; linear sieve; Selberg's $\Lambda ^2$-sieve 
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     title = {On the least almost-prime in arithmetic progressions},
     journal = {Czechoslovak Mathematical Journal},
     pages = {535--548},
     year = {2024},
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Wu, Liuying. On the least almost-prime in arithmetic progressions. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 535-548. doi: 10.21136/CMJ.2024.0459-23

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