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Integers free of large prime factors and the Riemann hypothesis

Published online by Cambridge University Press:  26 February 2010

Adolf Hildebrand
Adolf Hildebrand
Affiliation:
Department of Mathematics, University of Illinois, 1409, West Green Street, Urbana, Illinois 61801, U.S.A.
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For x, y ≥ 1, let Ψ(x, y) denote the number of positive integers less than or equal to x and free of prime factors greater than y. The behaviour of the function Ψ(x, y) has been the object of numerous articles (see e.g. Norton's memoir [5] and the bibliography there). It turns out that a good approximation to ψ(x, y)/x is given by ρ(log x/log y), where the function ρ(t) is defined for t ≥ 0 as the continuous solution of the equations

Type
Research Article
Information
Mathematika , Volume 31 , Issue 2 , December 1984 , pp. 258 - 271
Copyright
Copyright © University College London 1984

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References

1.Alladi, K.. The Turan-Kubilius inequality for integers without large prime factors. J. Reine Angew. Math., 335 (1982), 180–196.Google Scholar
2.de Bruijn, N. G.. On the number of positive integers ≥ x and free of prime factors > y. Ned. Akad. Wetensch. Proc. Ser. A., 54 (1951), 50–60.Google Scholar
3.Dickman, K.. On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astr. Fys., 22 (1930), 1–14.Google Scholar
4.Hildebrand, A.. On the number of positive integers ≥ x and free of prime factors ≥ y. J. Number Theory. To appear.Google Scholar
5.Norton, K.. Numbers with small prime factors and the least K-th power non-residue. Memoirs of the Amer. Math. Soc, 106 (1971), 1–106.Google Scholar