Note on a conjecture of Hildebrand regarding friable integers

Abstract

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