Czechoslovak Mathematical Journal, Vol. 73, No. 1, pp. 177-188, 2023
On the least almost-prime in arithmetic progression
Jinjiang Li, Min Zhang, Yingchun Cai
Received December 28, 2021. Published online October 31, 2022.
Abstract: 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.
Keywords: almost-prime; arithmetic progression; linear sieve; Selberg's $\Lambda^2$-sieve
Affiliations: Jinjiang Li, Department of Mathematics, China University of Mining and Technology, Beijing 100083, P. R. China, e-mail: jinjiang.li.math@gmail.com; Min Zhang (corresponding author), School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, P. R. China, e-mail: min.zhang.math@gmail.com; Yingchun Cai, School of Mathematical Science, Tongji University, Shanghai 200092, P. R. China, e-mail: yingchuncai@tongji.edu.cn
