# Order of magnitude for the number of y-smooth numbers less than or equal to x

Let $\Psi (x,y)$ denote the number of $y$-smooth integers less than or equal to $x$.

I know the precise results about $\Psi (x,y)$ in the literature (with good error terms), but what are some helpful "order of magnitudes" to keep in mind, that give a representation of what's going on when $y$ varies?

$$\Psi(x,y) = x \rho(u) \Big( 1 + O\Big(\frac{\log(u+1)}{\log y}\Big)\Big) \qquad (x \geq 3,\ e^{(\log_2x)^{5/3+\varepsilon}} \leq y \leq x)$$ (see §III.5.5 of [T15])
$$\Psi(x,y) \ll x e^{-u\log u} + \sqrt{x} \qquad (x \geq2,\ y \geq 2)$$ (see §III.5.6 of [T15]), we can get the following order of magnitudes for the number of $y$-smooth numbers up to $x$:
$$\begin{array}{c|c|c} y & \log y & u = \frac{\log x}{\log y} & \Psi(x,y) \\ \hline x & \log x & 1 & x\\ x^\alpha & \alpha \log x & \frac1\alpha & \sim \rho(1/\alpha) x \\ e^{\frac{(1-\varepsilon) (\log x) (\log_3x)}{\log_2 x}} & \frac{(1-\varepsilon)(\log x) (\log_3x)}{\log_2 x} & \frac{\log_2 x}{(1-\varepsilon)\log_3 x} & \ll \frac{x}{\log x} \\ e^{\frac{\log x}{\log_2 x}} & \frac{\log x }{\log_2 x} & \log_2 x & \ll \frac{x}{(\log x)^{\log_3 x}} \\ e^{\sqrt{\log x}} & \sqrt{\log x} & \sqrt{\log x} & \ll \frac{x}{(\log x)^{(1/2)\sqrt{\log x}}} \\ e^{(\log_2 x)^{5/3}} & (\log_2 x)^{5/3} & \frac{\log x}{(\log_2 x)^{5/3}} & \ll x^{1 - \frac{1-\varepsilon}{(\log_2 x)^{2/3}}}\\ (\log x)^{10} & 10\log_2 x & \frac{\log x}{10 \log_2 x} & \ll x^{0.9+\varepsilon} \\ (\log x)^2 & 2\log_2 x & \frac{\log x}{2 \log_2 x} & \ll x^{0.5+\varepsilon} \\ \log x & \log_2 x & \frac{\log x}{\log_2 x} & \ll \sqrt{x} \\ \end{array}$$