Note on a conjecture of Hildebrand regarding friable integers
Volume 208 / 2023
Acta Arithmetica 208 (2023), 279-283
MSC: Primary 11N25.
DOI: 10.4064/aa221127-24-4
Published online: 26 May 2023
Abstract
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.
