On the least almost-prime in arithmetic progressions

Liuying Wu

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 2, page 535-548
  • ISSN: 0011-4642

Abstract

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Let 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 𝒫 2 ( q , a ) denote the least 𝒫 2 in the arithmetic progression { n q + a } n = 1 ∞ . It is proved that for sufficiently large q , we have 𝒫 2 ( q , a ) ≪ q 1 . 825 . This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained 𝒫 2 ( q , a ) ≪ q 1 . 8345 .

How to cite

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Wu, Liuying. "On the least almost-prime in arithmetic progressions." Czechoslovak Mathematical Journal 74.2 (2024): 535-548. <http://eudml.org/doc/299549>.

@article{Wu2024,
abstract = {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 $\lbrace nq+a\rbrace _\{n=1\}^\{\infty \}$. 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 $\mathcal \{P\}_\{2\}(q,a)\ll q^\{1.8345\}.$},
author = {Wu, Liuying},
journal = {Czechoslovak Mathematical Journal},
keywords = {almost-prime; arithmetic progression; linear sieve; Selberg’s $\Lambda ^2$-sieve},
language = {eng},
number = {2},
pages = {535-548},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the least almost-prime in arithmetic progressions},
url = {http://eudml.org/doc/299549},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Wu, Liuying
TI - On the least almost-prime in arithmetic progressions
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 535
EP - 548
AB - 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 $\lbrace nq+a\rbrace _{n=1}^{\infty }$. 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 $\mathcal {P}_{2}(q,a)\ll q^{1.8345}.$
LA - eng
KW - almost-prime; arithmetic progression; linear sieve; Selberg’s $\Lambda ^2$-sieve
UR - http://eudml.org/doc/299549
ER -

References

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