Geodesic
On the least almost-prime in arithmetic progression
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 177-188
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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 $1.8345$.
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 $1.8345$.
DOI : 10.21136/CMJ.2022.0478-21
Classification : 11N13, 11N35, 11N36
Keywords: almost-prime; arithmetic progression; linear sieve; Selberg's $\Lambda ^2$-sieve 
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Li, Jinjiang; Zhang, Min; Cai, Yingchun. On the least almost-prime in arithmetic progression. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 177-188. doi: 10.21136/CMJ.2022.0478-21

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