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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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 $\Psi (x,y)$.
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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