Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1299–1326 | Cite as

Lois de répartition des diviseurs des entiers friables

Article
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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.

Keywords

Distribution of divisors Friable integers Saddle-point method Additive functions 

Mathematics Subject Classification

Primary 11N25 Secondary 11N37 11N60 

Notes

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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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Université d’Aix-Marseille, CNRS, Centrale Marseille, I2M UMR 7373MarseilleFrance
  2. 2.Institut Élie CartanUniversité de LorraineVandœuvre-lès-Nancy CedexFrance

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