×

Sur un problème de crible et ses applications. (On a sieve method and its applications). (French) Zbl 0599.10037

Let p(d) denote the least prime factor of d. The following function F was considered (implicitly) by A. Schinzel and G. Szekeres [Acta Sci. Math. 20, 221-229 (1959; Zbl 0099.027)]: \(F(n)=\max \{d p(d):\) d \(| n\), \(d>1\}\), \(n>1\), with \(F(1)=1\). In the spirit of the well-known linear sieve of Rosser and Iwaniec (in which, however, the factor p(d) would appear raised to a nonlinear power \(\{p(d)\}^{\beta})\) define \(E(x,y)=card\{n\leq x:\) F(n)\(\leq yx\}.\)
The principal result of the paper implies upper and lower bounds for \(E(x,x^{1/u})\) when \(u\geq 1\) and \(x^{1/u}\geq 2\). The upper bound is of order (x/u)log u but the lower bound is of order (x/u)\(\{\) (log u)\({}^{-\lambda}\}\) where \(\lambda\) is somewhat greater than 4 (though a better lower bound would follow from the Riemann hypothesis).
Three applications of this result are given. One of them depends on the identification \[ F(n)/n=\max_{1\leq i\leq d(n)}\{d_{i+1}/d_ i\}\quad (n>1) \] where \(1=d_ 1<d_ 2<...<d_{d(n)}=n\) is the increasing sequence of divisors of n. The second improves a result of M. Margenstern [C. R. Acad. Sci., Paris, Sér. I 299, 895-898 (1984; Zbl 0572.10007)] on practical numbers n (such that every \(m\leq n\) can be written as a sum of distinct divisors of n). Upper and lower bounds are obtained for the number of practical numbers not exceeding x.
The third relates to the ”small sieve” of P. Erdős and I. Z. Ruzsa [J. Number Theory 12, 385-394 (1980; Zbl 0435.10028)]. Let F(x,A) denote the number of integers \(\leq x\) not divisible by any element of a sequence A of integers. Define H(x,K) to be the minimum of F(x,A) taken over all A for which \(\sum_{a\in A}a^{-1}\leq K\). The author improves an upper estimate of I. Z. Ruzsa [ibid. 14, 260-268 (1982; Zbl 0481.10045)] for the quantity H(x,1).
Reviewer: G.Greaves

MSC:

11N35 Sieves
11N37 Asymptotic results on arithmetic functions
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] J. D. BOVEY , On the Size of Prime Factors of Integers (Acta Arith., vol. 33, 1977 , p. 65-80). Article | MR 437476 | Zbl 0353.10033 · Zbl 0353.10033
[2] P. ERDÖS , On the Distribution Function of Additive Functions (Ann. of Math., vol. 47, 1946 , p. 1-20). MR 15424 | Zbl 0061.07902 · Zbl 0061.07902 · doi:10.2307/1969031
[3] P. ERDÖS On Some Properties of Prime Factors of Integers (Nagoya Math. J., vol. 27, 1966 , p. 617-623). Article | MR 204378 | Zbl 0151.03501 · Zbl 0151.03501
[4] P. ERDÖS et I. Z. RUZSA , On the Small Sieve. I. Sifting by Primes , (J. Number Theory, vol. 12, 1980 , p. 385-394). MR 586468 | Zbl 0435.10028 · Zbl 0435.10028 · doi:10.1016/0022-314X(80)90032-3
[5] J. B. FRIEDLANDER , Integers Free from Large and Small Primes (Proc. London Math. Soc., (3), n^\circ 33, 1976 , p. 565-576). MR 417078 | Zbl 0344.10021 · Zbl 0344.10021 · doi:10.1112/plms/s3-33.3.565
[6] G. HALÁSZ , Remarks to my paper: ”On the Distribution of Additive and the Mean Value of Multiplicative Arithmetic Functions” (Acta Math. Acad. Scient. Hung., vol. 23, (3-4), 1972 , p. 425-432). MR 319931 | Zbl 0255.10046 · Zbl 0255.10046 · doi:10.1007/BF01896961
[7] H. HALBERSTAM et K. F. ROTH , Sequences, Oxford at the Clarendon Press, 1966 . MR 210679 | Zbl 0141.04405 · Zbl 0141.04405
[8] G. H. HARDY et E. M. WRIGHT , An Introduction to the Theory of Numbers , Oxford at the Clarendon Press, 5e éd., 1979 . MR 568909 | Zbl 0423.10001 · Zbl 0423.10001
[9] M. HAUSMAN et H. N. SHAPIRO , On Pratical Numbers (Comm. Pure and Applied Math., vol. 37, 1984 , p. 705-713). MR 752596 | Zbl 0544.10005 · Zbl 0544.10005 · doi:10.1002/cpa.3160370507
[10] M. N. HUXLEY , The distribution of prime numbers , Oxford at the Clarendon Press, 1972 . MR 444593 | Zbl 0248.10030 · Zbl 0248.10030
[11] H. IWANIEC , Rosser’s Sieve - Bilinear Forms of the Remainder Terms - Some Applications (Recent Progress in Analytic Number Theory, Vol. 1, H. HALBERSTAM and C. HOOLEY éd., Academic Press, 1981 , p. 203-230). MR 637348 | Zbl 0457.10026 · Zbl 0457.10026
[12] M. MARGENSTERN , Résultats et conjectures sur les nombres pratiques (C.R. Acad. Sc. Paris, t. 299, série I, n^\circ 18, 1984 , p. 895-898). MR 774662 | Zbl 0572.10007 · Zbl 0572.10007
[13] K. K. NORTON , On the Number of Restricted Prime Factors of an Integer I (Ill. J. Math., vol. 20, 1976 , p. 681-705). MR 419382 | Zbl 0329.10035 · Zbl 0329.10035
[14] I. Z. RUZSA , On the Small Sieve II. Sifting by Composite Numbers (J. Number Theory, vol. 14, 1982 , p. 260-268). MR 655730 | Zbl 0481.10045 · Zbl 0481.10045 · doi:10.1016/0022-314X(82)90051-8
[15] A. SCHINZEL et G. SZEKERES , Sur un problème de M. Paul Erdös (Acta Sc. Math. Szeged, vol. 20, 1959 , p. 221-229). MR 112864 | Zbl 0099.02702 · Zbl 0099.02702
[16] B. M. STEWART , Sums of Distinct Divisors (Amer. J. Math., vol. 76, 1954 , p. 779-785). MR 64800 | Zbl 0056.27004 · Zbl 0056.27004 · doi:10.2307/2372651
[17] G. TENENBAUM , Lois de répartition des diviseurs , 5 (J. London Math. Soc., (2), n^\circ 20, 1979 , p. 165-176). MR 551441 | Zbl 0422.10050 · Zbl 0422.10050 · doi:10.1112/jlms/s2-20.2.165
[18] G. TENENBAUM , Sur la probabilité qu’un entier possède un diviseur dans un intervalle donné (Compositio Math., vol. 51, 1984 , p. 243-263). Numdam | MR 739737 | Zbl 0541.10038 · Zbl 0541.10038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.