Let $\Psi(x,B)$ denote the number of $B$-smooth numbers less than $x$. [Wikipedia](https://en.wikipedia.org/wiki/Smooth_number#Distribution) gives the following "good estimate" for small, fixed $B$:


$$\Psi(x,B) \sim \frac{1}{\pi(B)!} \prod\_{p\le B}\frac{\log x}{\log p}$$


However, no citation or further discussion is given. Where does this estimate come from? How good an estimate is it? Anyone know a reference for this?



Estimate for de Bruijn function with small fixed smoothness bound
This is discussed in chapter III.5 of Tenenbaum *Intro to analytic and probabilistic number theory* (Cambridge, 1995). The estimate up to a factor $1+O(B^2/\log x\log B)$ uniform in $2 \le B \le \sqrt{\log x\log\log x}$ is attributed to Ennola 1969.


