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Lois de répartition des diviseurs des entiers friables

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Abstract

According to a general probabilistic principle, the natural divisors of friable integers (i.e. integers free of large prime factors) should normally present a Gaussian distribution. We show that this indeed is the case with conditional density tending to 1 as soon as the standard necessary conditions are met. Furthermore, we provide explicit, essentially optimal estimates for the decay of the involved error terms. The size of the exceptional set is sufficiently small to enable recovery of the average behaviour in the same optimal range. Our argument combines the saddle-point method with new large deviations estimates for the distribution of certain additive functions.

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Remerciement

Les auteurs tiennent ici à remercier chaleureusement l’arbitre pour la pertinence, la précision et la complétude de son rapport.

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Authors and Affiliations

  1. Université d’Aix-Marseille, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France

    Sary Drappeau

  2. Institut Élie Cartan, Université de Lorraine, BP 70239, 54506, Vandœuvre-lès-Nancy Cedex, France

    Gérald Tenenbaum

Authors
  1. Sary Drappeau
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  2. Gérald Tenenbaum
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Correspondence to Sary Drappeau.

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Drappeau, S., Tenenbaum, G. Lois de répartition des diviseurs des entiers friables. Math. Z. 288, 1299–1326 (2018). https://doi.org/10.1007/s00209-017-1935-7

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  • DOI: https://doi.org/10.1007/s00209-017-1935-7

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