On the least almost-prime in arithmetic progressions

Wu, Liuying

Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 535-548

Abstract (VO)

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 $\mathcal P_{2}(q,a)\ll q^{1.8345}.$

Détail

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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