## Sieving integers without large prime factors. (Cribler les entiers sans grand facteur premier.)(French)Zbl 0795.11042

Let $$P(n)$$ denote the largest prime factor of $$n$$, and $\Psi(x,y)= |\{n\leq x:\;P(n)\leq y\}|, \qquad \Psi_ q(x,y)= |\{n\leq x:\;P(n)\leq y,\;(n,q)=1\}|$ where $$q$$ is a positive integer. Various authors, including E. Fouvry and G. Tenenbaum in [Proc. Lond. Math. Soc., III. Ser. 63, 449-494 (1991; Zbl 0745.11042)], have considered the problem of determining conditions under which the asymptotic formula $\Psi_ q(x,y)\sim {\textstyle {{\phi(q)} \over q}} \Psi(x,y) \tag{*}$ holds. The object of the present paper is to establish (*) with an optimal error term in the case when $$P(q)\leq y\leq x$$ and $$\omega(q)\leq y^{c/\log(1+u)}$$ with $$u= {{\log x} \over {\log y}}$$ and $$c$$ an arbitrary positive constant. In a useful list of remarks, the author describes how his result relates to those in other recent papers. The proof utilizes key auxiliary functions that have been featured previously in the literature.

### MSC:

 11N25 Distribution of integers with specified multiplicative constraints 11N36 Applications of sieve methods

Zbl 0745.11042
Full Text: