For an asymptotic estimate for the count of $y$-smooth numbers between $0$ and $x$, with the Dickman-de Bruijn function $\Psi(x, y) \approx x \rho(u)$ where $u = log( x)/log( y)$, Hildebrand and Tenenbaum provides the simple bounds for $\rho(u)$:
$0 < \rho(u) < 1/ \Gamma(u + 1)$.
Does an upper bound for $\rho(u)$ necessarily imply that $x/\Gamma(a+1)$ is an upper bound for $\Psi(x, y)$? My preliminary data suggests that this is true. If so, is it because $x\rho(u)$ is asymptotic that makes this true?
If not, I think I have some brute force checks to do to compare this function versus known upper bound formulas.