Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-20T00:38:51.523Z Has data issue: false hasContentIssue false
  • English
  • Français

On Behrend sequences

Published online by Cambridge University Press:  24 October 2008

R. R. Hall  and
G. Tenenbaum
R. R. Hall
Affiliation:
Department of Mathematics, York University, Heslington, York YOl 5DD, England
G. Tenenbaum
Affiliation:
Département de Mathématiques, Université de Nancy I, BP 239, 54506 Vandoeuvre Cedex, France
Get access Rights & Permissions [Opens in a new window]

Extract

Let denote a sequence of integers exceeding 1, and let τ(n, ) be the number of those divisors of n which belong to . We say that is a Behrend sequence if

where, here and in the sequel, we use the notation p.p. to indicate that a relation holds on a set of asymptotic density one.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 3 , November 1992 , pp. 467 - 482
Copyright
Copyright © Cambridge Philosophical Society 1992

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]Behrend, F. A.. Generalizations of an inequality of Heilbronn and Rohrbach. Bull. Amer. Math. Soc. 54 (1948), 681–684.CrossRefGoogle Scholar
[2]Davenport, H. and Erdõs, P.. On sequences of positive integers. Acta Arith. 2 (1937), 147–151.CrossRefGoogle Scholar
[3]Davenport, H. and Erdõs, P.. On sequences of positive integers. J. Indian Math. Soc. 15 (1951), 19–24.Google Scholar
[4]Erdõs, P.. Some remarks on prime factors of integers. Canad. J. Math. 11 (1959), 161–167.CrossRefGoogle Scholar
[5]Erdõs, P.. Some unconventional problems in number theory. Astérisque 61 (1979), 73–82.Google Scholar
[6]Erdõs, P. and Hall, R. R.. The propinquity of divisors. Bull. London Math. Soc. 11 (1979), 304–307.CrossRefGoogle Scholar
[7]Halberstam, H. and Richest, H.-E.. On a result of R. R. Hall. J. Number Theory 11, 76–89.CrossRefGoogle Scholar
[8]Hall, R. R.. Sets of multiples and Behrend sequences. In A Tribute to Paul Erdos (editors Baker, A., Bollobas, B., Hajnal, A.), (Cambridge University Press, 1990), pp. 249–258.CrossRefGoogle Scholar
[9]Hall, R. R. and Tenenbaum, G.. Les ensembles de multiples et la densite divisorielle. J. Number Theory 22 (1986), 308–333.CrossRefGoogle Scholar
[10]Hall, R. R. and Tenenbaum, G.. Divisors (Cambridge University Press, 1988).CrossRefGoogle Scholar
[11]Maier, H. and Tenenbaum, G.. On the set of divisors of an integer. Invent. Math. 76 (1984), 121–128.CrossRefGoogle Scholar
[12]Tenenbaum, G.. Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné. Compositio Math. 51 (1984), 243–263.Google Scholar
[13]Tenenbaum, G.. Introduction à la Théorie Analytique et Probabiliste des Nombres. Revue de l'lnstitut Elie Cartan 13, Département de Mathématiques de l'université de Nancy I (1990).Google Scholar