On the least almost-prime in arithmetic progression

Li, Jinjiang, Zhang, Min, and Cai, Yingchun. "On the least almost-prime in arithmetic progression." Czechoslovak Mathematical Journal 73.1 (2023): 177-188. <http://eudml.org/doc/299370>.

@article{Li2023,
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 $\mathcal \{P\}_2\equiv a\hspace\{4.44443pt\}(\@mod \; q)$. 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 $1.8345$.},
author = {Li, Jinjiang, Zhang, Min, Cai, Yingchun},
journal = {Czechoslovak Mathematical Journal},
keywords = {almost-prime; arithmetic progression; linear sieve; Selberg’s $\Lambda ^2$-sieve},
language = {eng},
number = {1},
pages = {177-188},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the least almost-prime in arithmetic progression},
url = {http://eudml.org/doc/299370},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Li, Jinjiang
AU - Zhang, Min
AU - Cai, Yingchun
TI - On the least almost-prime in arithmetic progression
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 1
SP - 177
EP - 188
AB - 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 $\mathcal {P}_2\equiv a\hspace{4.44443pt}(\@mod \; q)$. 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 $1.8345$.
LA - eng
KW - almost-prime; arithmetic progression; linear sieve; Selberg’s $\Lambda ^2$-sieve
UR - http://eudml.org/doc/299370
ER -