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?



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


$$\Psi(x,y) = x \rho(u) \Big( 1 + O\Big(\frac{\log(u+1)}{\log y}\Big)\Big) \qquad (x \geq 3,\ e^{(\log\_2x)^{5/3+\varepsilon}} \leq y \leq x)$$ (see §III.5.5 of [T15])


or 


$$\Psi(x,y) \ll x e^{-u\log u} + \sqrt{x} \qquad (x \geq2,\ y \geq 2)$$ (see §III.5.6 of [T15]), we can get the following order of magnitudes for the number of $y$-smooth numbers up to $x$:


$$
\begin{array}{c|c|c}
y & \log y & u = \frac{\log x}{\log y} & \Psi(x,y) \\
\hline
x & \log x & 1 & x\\
x^\alpha & \alpha \log x & \frac1\alpha & \sim \rho(1/\alpha) x \\
e^{\frac{(1-\varepsilon) (\log x) (\log\_3x)}{\log\_2 x}} & \frac{(1-\varepsilon)(\log x) (\log\_3x)}{\log\_2 x} & \frac{\log\_2 x}{(1-\varepsilon)\log\_3 x} & \ll \frac{x}{\log x} \\
e^{\frac{\log x}{\log\_2 x}} & \frac{\log x }{\log\_2 x} & \log\_2 x & \ll \frac{x}{(\log x)^{\log\_3 x}} \\
e^{\sqrt{\log x}} & \sqrt{\log x} & \sqrt{\log x} & \ll \frac{x}{(\log x)^{(1/2)\sqrt{\log x}}} \\
e^{(\log\_2 x)^{5/3}} & (\log\_2 x)^{5/3} & \frac{\log x}{(\log\_2 x)^{5/3}} & \ll x^{1 - \frac{1-\varepsilon}{(\log\_2 x)^{2/3}}}\\
(\log x)^{10} & 10\log\_2 x & \frac{\log x}{10 \log\_2 x} & \ll x^{0.9+\varepsilon} \\
(\log x)^2 & 2\log\_2 x & \frac{\log x}{2 \log\_2 x} & \ll x^{0.5+\varepsilon} \\
\log x & \log\_2 x & \frac{\log x}{\log\_2 x} & \ll \sqrt{x} \\
\end{array} 
$$


[T15] G. Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd edition, AMS, 2015.


