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Transactions of the American Mathematical Society

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



Sieving the positive integers by small primes

Authors: D. A. Goldston and Kevin S. McCurley
Journal: Trans. Amer. Math. Soc. 307 (1988), 51-62
MSC: Primary 11N35
MathSciNet review: 936804
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Abstract: Let $ Q$ be a set of primes that has relative density $ \delta $ among the primes, and let $ \phi (x,\,y,\,Q)$ be the number of positive integers $ \leqslant x$ that have no prime factor $ \leqslant y$ from the set $ Q$. Standard sieve methods do not seem to give an asymptotic formula for $ \phi (x,\,y,\,Q)$ in the case that $ \tfrac{1}{2} \leqslant \delta < 1$. We use a method of Hildebrand to prove that

$\displaystyle \phi (x,y,Q)\tilde{x}f(u)\prod\limits_{\mathop {p < y}\limits_{p \in Q} } {\left( {1 - \frac{1}{p}} \right)} $

as $ x \to \infty $, where $ u = \frac{{\log x}}{{\log y}}$ and $ f(u)$ is defined by

$\displaystyle {u^\delta }f(u) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{e^{{\... ...1 + t)}^{\delta - 1}}\;dt,} } \hfill & {u > 1.} \hfill \\ \end{array} } \right.$

This may also be viewed as a generalization of work by Buchstab and de Bruijn, who considered the case where $ Q$ consisted of all primes.

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Article copyright: © Copyright 1988 American Mathematical Society

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