Note on a conjecture of Hildebrand regarding friable integers

TL;DR

This paper provides a concise proof confirming Hildebrand's conjecture on the behavior of the count of y-friable integers, establishing the boundary where the smooth approximation holds or fails.

Contribution

The paper offers a simplified, direct proof of Gorodetsky's recent confirmation of Hildebrand's conjecture regarding friable integers.

Findings

Confirmed the conjecture for y > (log x)^{2+ε}

Established the failure of the approximation for y ≤ (log x)^{2−ε}

Provided a shorter proof of Gorodetsky's result

Abstract

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