On integers free of large prime factors

Hildebrand, Adolf; Tenenbaum, Gérald

doi:10.1090/S0002-9947-1986-0837811-1

On integers free of large prime factors
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by Adolf Hildebrand and Gérald Tenenbaum PDF

Trans. Amer. Math. Soc. 296 (1986), 265-290 Request permission

Abstract:

The number $\Psi (x,y)$ of integers $\leq x$ and free of prime factors $> y$ has been given satisfactory estimates in the regions $y \leq {(\log x)^{3/4 - \varepsilon }}$ and $y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\}$. In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates $\Psi (x,y)$ uniformly for $x \geq y \geq 2$ within a factor $1 + O((\log y)/(\log x) + (\log y)/y)$. As an application, we derive a simple formula for $\Psi (cx,y)/\Psi (x,y)$, where $1 \leq c \leq y$. We also prove a short interval estimate for $\Psi (x,y)$.

References

Krishnaswami Alladi, The Turán-Kubilius inequality for integers without large prime factors, J. Reine Angew. Math. 335 (1982), 180–196. MR 667466, DOI 10.1515/crll.1982.335.180
N. G. de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.) 15 (1951), 25–32. MR 43838
N. G. de Bruijn, On the number of positive integers $\leq x$ and free of prime factors $>y$, Nederl. Acad. Wetensch. Proc. Ser. A. 54 (1951), 50–60. MR 0046375
N. G. de Bruijn, On the number of positive integers $\leq x$ and free prime factors $>y$. II, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math. 28 (1966), 239–247. MR 0205945
E. Rodney Canfield, The asymptotic behavior of the Dickman-de Bruijn function, Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982), 1982, pp. 139–148. MR 725875
E. R. Canfield, Paul Erdős, and Carl Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), no. 1, 1–28. MR 712964, DOI 10.1016/0022-314X(83)90002-1

On the frequency of numbers containing prime factors of a certain relative magnitude

22

William John Ellison, Les nombres premiers, Publications de l’Institut de Mathématique de l’Université de Nancago, No. IX, Hermann, Paris, 1975 (French). En collaboration avec Michel Mendès France. MR 0417077
Veikko Ennola, On numbers with small prime divisors, Ann. Acad. Sci. Fenn. Ser. A I No. 440 (1969), 16. MR 0244175
Douglas Hensley, The number of positive integers $\leq x$ and free of prime factors $>y$, J. Number Theory 21 (1985), no. 3, 286–298. MR 814007, DOI 10.1016/0022-314X(85)90057-5
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, DOI 10.1016/0022-314X(86)90030-2
Adolf Hildebrand, On the number of positive integers $\leq x$ and free of prime factors $>y$, J. Number Theory 22 (1986), no. 3, 289–307. MR 831874, DOI 10.1016/0022-314X(86)90013-2

On the local behavior of

On integers free of large prime divisors

Karl K. Norton, Numbers with small prime factors, and the least $k$th power non-residue, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR 0286739
Karl Prachar, Primzahlverteilung, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0087685

The difference between consecutive prime numbers

13

A. I. Vinogradov, On numbers with small prime divisors, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 683–686 (Russian). MR 0085284