On integers free of large prime factors

Hildebrand, Adolf; Tenenbaum, Gérald

doi:10.1090/S0002-9947-1986-0837811-1

On integers free of large prime factors

Authors: Adolf Hildebrand and Gérald Tenenbaum
Journal: Trans. Amer. Math. Soc. 296 (1986), 265-290
MSC: Primary 11N25
DOI: https://doi.org/10.1090/S0002-9947-1986-0837811-1
MathSciNet review: 837811
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Abstract: The number $\Psi (x,y)$ of integers $\leq x$ and free of prime factors $> y$ has been given satisfactory estimates in the regions $y \leq {(\log x)^{3/4 - \varepsilon }}$ and $y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\}$. In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates $\Psi (x,y)$ uniformly for $x \geq y \geq 2$ within a factor $1 + O((\log y)/(\log x) + (\log y)/y)$. As an application, we derive a simple formula for $\Psi (cx,y)/\Psi (x,y)$, where $1 \leq c \leq y$. We also prove a short interval estimate for $\Psi (x,y)$.

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