Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T05:47:19.540Z Has data issue: false hasContentIssue false
  • English
  • Français

On the normal concentration of divisors, 2

Published online by Cambridge University Press:  06 July 2009

HELMUT MAIER  and
GÉRALD TENENBAUM
HELMUT MAIER
Affiliation:
Universitát Ulm, Abt. Mathematik III, Helmholtzstrasse 18, D-89069 Ulm, Germany
GÉRALD TENENBAUM
Affiliation:
Institut Élie Cartan, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes, BP 239, 54506 Vandœuvre Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We improve the current upper and lower bounds for the normal order of the Erdős–Hooley Δ–functionobtaining, for almost all integers n, the inequalitieswhere the exponent γ := (log 2)/log((1−1/log 27)/(1 − 1/log 3)) ≈ 0.33827 is conjectured to be optimal.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 3 , November 2009 , pp. 513 - 540
Copyright
Copyright © Cambridge Philosophical Society 2009

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

REFERENCES

[1]Erdős, P.On the density of some sequences of integers. Bull. Amer. Math. Soc. 54 (1948), 685–692.CrossRefGoogle Scholar
[2]Erdős, P.Problem 218 and solution. Canad. Math. Bull. 17 (1974), 621–622.Google Scholar
[3]Erdős, P. and Tenenbaum, G.Sur les fonctions arithmétiques liées aux diviseurs consécutifs. J. Number Theory, 31 (1989), 285–311.CrossRefGoogle Scholar
[4]Hall, R.R. and Tenenbaum, G.On the average and normal orders of Hooley's Δ-function. J. London Math. Soc (2) 25 (1982), 392–406.CrossRefGoogle Scholar
[5]Hall, R.R. and Tenenbaum, G.Divisors. Cambridge tracts in mathematics 90. (Cambridge University Press (1988), (paperback ed. 2008).Google Scholar
[6]Hooley, C.A new technique and its applications to the theory of numbers. Proc. London Math. Soc. (3) 38 (1979), 115–151.CrossRefGoogle Scholar
[7]Maier, H. and Tenenbaum, G.On the set of divisors of an integer. Invent. Math. 76 (1984), 121–128.Google Scholar
[8]Maier, H. and Tenenbaum, G.On the normal concentration of divisors. J. London Math. Soc. (2) 31 (1985), 393–400.Google Scholar
[9]Raouj, A. and Tenenbaum, G.Sur l'écart quadratique moyen des diviseurs d'un entier normal, Math. Proc. Camb. Phil. Soc. 126 (1999), 399–415.Google Scholar
[10]Shiu, P., A Brun–Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313 (1980), 161–170.Google Scholar
[11]Tenenbaum, G.Sur la concentration moyenne des diviseurs. Comment. Math. Helv. 60 (1985), 411–428.CrossRefGoogle Scholar
[12]Tenenbaum, G.Fonctions Δ de Hooley et applications. Séminaire de Théorie des nombres, Paris 1984-85. Prog. Math. 63 (1986), 225–239.Google Scholar
[13]Tenenbaum, G.Introduction to analytic and probabilistic number theory, Cambridge studies in Advanced Mathematics, no. 46, (Cambridge University Press, 1995).Google Scholar
[14]Tenenbaum, G.Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2. Math. Proc. Camb. Phil. Soc. 138 (2005), 1–8.CrossRefGoogle Scholar
[15]Tenenbaum, G. in collaboration with Jie Wu. Exercices corrigés de théorie analytique et probabiliste des nombres. Cours spécialisés, no 2 (Société Mathématique de France, 1996).Google Scholar