## On integers free of large prime factors

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- by Adolf Hildebrand and Gérald Tenenbaum PDF
- Trans. Amer. Math. Soc.
**296**(1986), 265-290 Request permission

## 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 $\Psi (x,y)$.## References

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## Additional Information

- © Copyright 1986 American Mathematical Society
- 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