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On block Behrend sequences

Profile image of Gérald TenenbaumGérald Tenenbaum

1996, Mathematical Proceedings of the Cambridge Philosophical Society

https://doi.org/10.1017/S0305004100074910
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Abstract
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This work investigates block Behrend sequences, defined as strictly increasing sequences of integers whose set of multiples exhibits asymptotic density 1. The study extends classical results regarding coprime sequences and develops comprehensive criteria for a broader class of sequences. Through a careful exploration of block sequences characterized by specific growth conditions and statistical properties, the analysis aims at providing necessary and sufficient conditions for the Behrend property, contributing to the understanding of arithmetic properties in number theory.

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References (12)

  1. H. Davenport & P. Erdős, On sequences of positive integers, Acta Arith. 2 (1937), 147-151.
  2. H. Davenport & P. Erdős, On sequences of positive integers, J. Indian Math. Soc. 15 (1951), 19-24.
  3. P. Erdős, Some unconventional problems in number theory, Astérisque 61 (1979), 73-82.
  4. P. Erdős, R.R Hall & G. Tenenbaum, On the densities of sets of multiples, J. reine angew. Math. 454 (1994), 119-141.
  5. H. Halberstam & H.-E. Richert, Sieve methods, Academic Press, London, New York, San Franscisco, 1974.
  6. R.R Hall & G. Tenenbaum, Divisors, Cambridge University Press, 1988.
  7. R.R Hall & G. Tenenbaum, On Behrend sequences, Math. Proc. Camb. Phil. Soc. 112 (1992), 467-482.
  8. H. Maier & G. Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984), 121-128.
  9. I. Ruzsa & G. Tenenbaum, A note on Behrend sequences, Acta Math. Hung., to appear.
  10. P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. reine angew. Math. 313 (1980), 161-170.
  11. G. Tenenbaum, Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné, Compositio Math. 51 (1984), 243-263.
  12. G. Tenenbaum, Uniform distribution on divisors and Behrend sequences, preprint.

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Related topics

  • Mathematics
  • Pure Mathematics

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    1. We shall make use of this extra information later on. 2. See [29] for a short proof.

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