Sur un problème extrémal en arithmétique. (On an extremal problem in arithmetic). (French) Zbl 0622.10030

Tenenbaum, Gérald

Ann. Inst. Fourier 37, No. 2, 1-18 (1987); corrigendum ibid. 50, No. 1, 317-319 (2000).

Let \(F_{\alpha}(n)=\sum (d_{i+1}/d_ i-1)^{\alpha}\), where \(d_ i\) runs through the sequence of divisors of n in their natural order. Erdős had conjectured that, for any fixed \(\alpha >1\), (i) \(F_{\alpha}(n)\) is bounded on an infinite sequence of values n, and (ii) \(F_{\alpha}(n)\) is bounded uniformly for all values n of the form \(n=k!\), \(n=[1,2,...,k]\) and \(n=p_ 1...p_ k,\) where \(p_ i\) denotes the i-th prime. (i) was recently proved by M. D. Vose [J. Number Theory 19, 233-238 (1984; Zbl 0543.10031)].
In the present paper the author proves Erdős’ conjecture in its strong version (ii), and in fact a generalization of it. The proof is different from Vose’s. The author first reduces the problem to that of estimating the number of divisors of n in short intervals near \(\sqrt{n}\). He then applies Fourier techniques to obtain the required estimates for integers n that are of the above form.

Reviewer: A.Hildebrand

MSC:

11N37	Asymptotic results on arithmetic functions
11K65	Arithmetic functions in probabilistic number theory
11A05	Multiplicative structure; Euclidean algorithm; greatest common divisors

Keywords:

multiplicative structure of integers; arithmetic functions; consecutive divisors in small ratio

Citations:

Zbl 0543.10031

References:

[1]	[1] et , Les nombres premiers, Hermann, Paris (1975). · Zbl 0313.10001
[2]	[2] , Some problems and results on additive and multiplicative number theory, Analytic Number Theory (Philadelphia, 1980), Lecture Notes 899 (1981), 171-182. · Zbl 0472.10002
[3]	[3] et , Sur la structure de la suite des diviseurs d’un entier, Ann. Inst. Fourier, 31-1 (1981), 17-37. · Zbl 0437.10020
[4]	[4] et , An introduction to the theory of numbers, 5e éd., Oxford at the Clarendon Press (1979). · Zbl 0423.10001
[5]	[5] et , On integers free of large prime factors, Trans. Amer. Math. Soc., 296 (1986), 265-290. · Zbl 0601.10028
[6]	[6] , Primzahlverteilung, Springer, Berlin (1957). · Zbl 0080.25901
[7]	[7] , Majoration des dérivées des polynômes de Tchebychev, Communication privée (janvier 1983).
[8]	[8] , Integers with consecutive divisors in small ratio, J. Number Theory, 19 (1984), 233-238. · Zbl 0543.10031