Xuan, Tizuo Integers with no large prime factors. (English) Zbl 0819.11035 Acta Arith. 69, No. 4, 303-327 (1995). For real numbers \(x,y\geq 2\), let \(\Psi (x,y)\) denote the number of integers not exceeding \(x\) all of whose prime factors do not exceed \(y\), and let \(\Psi_ q (x,y)\) denote the number of such integers that are also coprime to \(q\). Also let \(\omega (n)\) denote the number of distinct prime factors of \(n\) and let \(q_ y\) denote the product of the prime divisors of \(q\) that are \(\leq y\). The asymptotic behaviour for \(\Psi_ q(x,y)\) has been studied by several authors, including K. K. Norton, D. G. Hazlewood, E. Fouvry and G. Tenenbaum, the author (unpublished) and improved recently by G. Tenenbaum [Philos. Trans. R. Soc. Lond., Ser. A 345, 377-384 (1993; Zbl 0795.11042)]. The following result is proved: the estimate \[ \Psi_ q (x,y)= \prod_{p\mid q, p\leq y} (1- p^{-\beta})\;\Psi (x,y) \Biggl\{ 1+O \biggl( {{\log (\omega(q_ y)+ 3)} \over {\log (u+1)\log y}} \biggr)+ O\left( \exp(- (\log y)^{3/5- \varepsilon} \bigr) \right)\Biggr\} \] holds uniformly for \(x\geq x_ 0\), \((\log x)^{1+ \varepsilon}\leq y\leq x\) and \(\omega (q_ y)\leq y^{1/2}\), where \(u=\log x/\log y\), \(\beta= \beta(x, y)= 1-\xi (u)/\log y\), and here \(\xi= \xi(u)\) for \(u>1\) is the positive solution of the equation \(e^ \varepsilon= u\xi +1\). Reviewer: Xuan Tizuo (Beijing) Cited in 2 Documents MSC: 11N25 Distribution of integers with specified multiplicative constraints 11N36 Applications of sieve methods Keywords:integers free of large prime factors; asymptotic estimates; sieve methods Citations:Zbl 0795.11042 PDFBibTeX XMLCite \textit{T. Xuan}, Acta Arith. 69, No. 4, 303--327 (1995; Zbl 0819.11035) Full Text: DOI EuDML