Sur un problème extrémal en arithmétique. (On an extremal problem in arithmetic)
type: Item
instance of: scholarly article
title: Sur un problème extrémal en arithmétique. (On an extremal problem in arithmetic)
zbMATH Open document ID: 0622.10030
author: Gérald Tenenbaum
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75418
review text: 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 \textit{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.
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11A05
zbMATH DE Number: 4008498
zbMATH Keywords: multiplicative structure of integers
zbMATH Keywords: arithmetic functions
zbMATH Keywords: consecutive divisors in small ratio
reviewed by: Adolf Hildebrand
MaRDI profile type: Publication
OpenAlex ID: W2318123129
cites work: Q4071794
cites work: Q3926660
cites work: Sur la structure de la suite des diviseurs d'un entier. (On the structure of the séquence of divisors of an integer)
cites work: Q3856819
cites work: On Integers Free of Large Prime Factors
cites work: Q3248054
cites work: Integers with consecutive divisors in small ratio
DOI: 10.5802/AIF.1083
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 4008498