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Transactions of the American Mathematical Society

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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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DOI: https://doi.org/10.1090/S0002-9947-1986-0837811-1
Article copyright: © Copyright 1986 American Mathematical Society

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