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Note on a conjecture of Hildebrand regarding friable integers

Volume 208 / 2023

Régis de la Bretèche, Gérald Tenenbaum Acta Arithmetica 208 (2023), 279-283 MSC: Primary 11N25. DOI: 10.4064/aa221127-24-4 Published online: 26 May 2023


Hildebrand proved that the smooth approximation for the number $\varPsi (x,y)$ of $y$-friable integers not exceeding $x$ holds for $y \gt (\log x)^{2+\varepsilon }$ under the Riemann hypothesis and he conjectured that it fails when $y\leqslant (\log x)^{2-\varepsilon }$. This conjecture has recently been confirmed by Gorodetsky by an intricate argument. We propose a short, straightforward proof.


  • Régis de la BretècheUniversité Paris Cité, Sorbonne Université, CNRS
    Institut de Math. de Jussieu – Paris Rive Gauche
    F-75013 Paris, France
  • Gérald TenenbaumInstitut Élie Cartan
    Université de Lorraine
    BP 70239
    54506 Vandœuvre-lès-Nancy Cedex, France

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