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Transactions of the American Mathematical Society

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

 

 

On the local behavior of $ \Psi(x,y)$


Author: Adolf Hildebrand
Journal: Trans. Amer. Math. Soc. 297 (1986), 729-751
MSC: Primary 11N25
DOI: https://doi.org/10.1090/S0002-9947-1986-0854096-0
MathSciNet review: 854096
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Abstract: $ \Psi (x,y)$ denotes the number of positive integers $ \leqslant x$ and free of prime factors $ > y$. In the range $ y \geqslant \exp ({(\log \log x)^{5/3 + \varepsilon }})$, $ \Psi (x,y)$ can be well approximated by a "smooth" function, but for $ y \leqslant {(\log x)^{2 - \varepsilon }}$, this is no longer the case, since then the influence of irregularities in the distribution of primes becomes apparent. We show that $ \Psi (x,y)$ behaves "locally" more regular by giving a sharp estimate for $ \Psi (cx,y)/\Psi (x,y)$, valid in the range $ x \geqslant y \geqslant 4\log x$, $ 1 \leqslant c \leqslant y$.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0854096-0
Keywords: Integers free of large primes factors, asymptotic estimates
Article copyright: © Copyright 1986 American Mathematical Society