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Integers with no large prime factors. (English) Zbl 0819.11035

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has been studied by several authors, including K. K. Norton, D. G. Hazlewood, E. Fouvry and G. Tenenbaum, the author (unpublished) and improved recently by G. Tenenbaum [Philos.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 \(e^ \varepsilon= u\xi +1\).

MSC:

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

Citations:

Zbl 0795.11042
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