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
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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  • [1] Krishnaswami Alladi, The Turán-Kubilius inequality for integers without large prime factors, J. Reine Angew. Math. 335 (1982), 180–196. MR 667466, 10.1515/crll.1982.335.180
  • [2] N. G. de Bruijn, On some Volterra equations of which all solutions are convergent, Indag. Math. 12 (1950), 257-265.
  • [3] -, On the number of positive integers $ \leqslant x$ and free of prime factors $ > y$, Indag. Math. 12 (1951), 50-60.
  • [4] K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys. 22 (1930), 1-14.
  • [5] D. R. Heath-Brown, Finding primes by sieve methods, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 487–492. MR 804704
  • [6] Douglas Hensley, A property of the counting function of integers with no large prime factors, J. Number Theory 22 (1986), no. 1, 46–74. MR 821136, 10.1016/0022-314X(86)90030-2
  • [7] Adolf Hildebrand, On the number of positive integers ≤𝑥 and free of prime factors >𝑦, J. Number Theory 22 (1986), no. 3, 289–307. MR 831874, 10.1016/0022-314X(86)90013-2
  • [8] Karl K. Norton, Numbers with small prime factors, and the least 𝑘th power non-residue, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR 0286739

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Keywords: Integers free of large primes factors, asymptotic estimates
Article copyright: © Copyright 1986 American Mathematical Society