Sums of powers: an arithmetic refinement to the probabilistic model of Erdős and Rényi

Jean-Marc Deshouillers; François Hennecart; Bernard Landreau

Acta Arithmetica (1998)

  • Volume: 85, Issue: 1, page 13-33
  • ISSN: 0065-1036

Abstract

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Erdős and Rényi proposed in 1960 a probabilistic model for sums of s integral sth powers. Their model leads almost surely to a positive density for sums of s pseudo sth powers, which does not reflect the case of sums of two squares. We refine their model by adding arithmetical considerations and show that our model is in accordance with a zero density for sums of two pseudo-squares and a positive density for sums of s pseudo sth powers when s ≥ 3. Moreover, our approach supports a conjecture of Hooley on the average of the square of the number of representations.

How to cite

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Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau. "Sums of powers: an arithmetic refinement to the probabilistic model of Erdős and Rényi." Acta Arithmetica 85.1 (1998): 13-33. <http://eudml.org/doc/207151>.

@article{Jean1998,
abstract = {Erdős and Rényi proposed in 1960 a probabilistic model for sums of s integral sth powers. Their model leads almost surely to a positive density for sums of s pseudo sth powers, which does not reflect the case of sums of two squares. We refine their model by adding arithmetical considerations and show that our model is in accordance with a zero density for sums of two pseudo-squares and a positive density for sums of s pseudo sth powers when s ≥ 3. Moreover, our approach supports a conjecture of Hooley on the average of the square of the number of representations.},
author = {Jean-Marc Deshouillers, François Hennecart, Bernard Landreau},
journal = {Acta Arithmetica},
keywords = {pseudo-squares; pseudo-powers; probabilistic model; sums of powers; arithmetic progressions; Poisson law; density; distribution},
language = {eng},
number = {1},
pages = {13-33},
title = {Sums of powers: an arithmetic refinement to the probabilistic model of Erdős and Rényi},
url = {http://eudml.org/doc/207151},
volume = {85},
year = {1998},
}

TY - JOUR
AU - Jean-Marc Deshouillers
AU - François Hennecart
AU - Bernard Landreau
TI - Sums of powers: an arithmetic refinement to the probabilistic model of Erdős and Rényi
JO - Acta Arithmetica
PY - 1998
VL - 85
IS - 1
SP - 13
EP - 33
AB - Erdős and Rényi proposed in 1960 a probabilistic model for sums of s integral sth powers. Their model leads almost surely to a positive density for sums of s pseudo sth powers, which does not reflect the case of sums of two squares. We refine their model by adding arithmetical considerations and show that our model is in accordance with a zero density for sums of two pseudo-squares and a positive density for sums of s pseudo sth powers when s ≥ 3. Moreover, our approach supports a conjecture of Hooley on the average of the square of the number of representations.
LA - eng
KW - pseudo-squares; pseudo-powers; probabilistic model; sums of powers; arithmetic progressions; Poisson law; density; distribution
UR - http://eudml.org/doc/207151
ER -

References

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  1. [1] R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., Providence, 1963. Zbl0128.04303
  2. [2] P. Barrucand, Sur la distribution empirique des sommes de trois cubes ou de quatre bicarrés, C. R. Acad. Sci. Paris A 267 (1968), 409-411. Zbl0162.06403
  3. [3] H. Davenport, Multiplicative Number Theory, Markham, 1967. 
  4. [4] P. Erdős and A. Rényi, Additive properties of random sequences of positive integers, Acta Arith. 6 (1960), 83-110. Zbl0091.04401
  5. [5] J. H. Goguel, Über Summen von zufälligen Folgen natürlichen Zahlen, J. Reine Angew. Math. 278/279 (1975), 63-77. 
  6. [6] H. Halberstam and R. F. Roth, Sequences, Clarendon Press, Oxford, 1966. 
  7. [7] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, 1985. Zbl0020.29201
  8. [8] C. Hooley, On some topics connected with Waring's problem, J. Reine Angew. Math. 369 (1986), 110-153. Zbl0589.10052
  9. [9] E. Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. Math. Phys. (3) 13 (1908), 305-312. 
  10. [10] B. Landreau, Modèle probabiliste pour les sommes de s puissances s-ièmes, Compositio Math. 99 (1995), 1-31. 
  11. [11] R. C. Vaughan, The Hardy-Littlewood Method, Cambridge Univ. Press, 1981. Zbl0455.10034

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