Hostname: page-component-84b7d79bbc-g7rbq Total loading time: 0 Render date: 2024-07-30T06:38:15.076Z Has data issue: false hasContentIssue false

Integers free of large prime factors and the Riemann hypothesis

Published online by Cambridge University Press:  26 February 2010

Adolf Hildebrand
Affiliation:
Department of Mathematics, University of Illinois, 1409, West Green Street, Urbana, Illinois 61801, U.S.A.
Get access

Extract

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
Copyright
Copyright © University College London 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alladi, K.. The Turan-Kubilius inequality for integers without large prime factors. J. Reine Angew. Math., 335 (1982), 180196.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), 5060.Google Scholar
3.Dickman, K.. On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astr. Fys., 22 (1930), 114.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), 1106.Google Scholar