Let $\Psi(x,B)$ denote the number of $B$-smooth numbers less than $x$. Wikipedia 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?

  • 2
    $\begingroup$ This is just from looking at the volume of $\sum_{p\le B} x_p \log p \le \log x$ with $x_p \ge 0$. Look in Granville's survey on smooth numbers for example. $\endgroup$ – Lucia Dec 12 '17 at 7:17

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.

  • $\begingroup$ Any info on what happens for larger B? I'm interested in B closer to log(x) $\endgroup$ – Elliot Gorokhovsky Dec 12 '17 at 21:04
  • 1
    $\begingroup$ The distribution for any range is explained both in Tenenbaum and Granville. $\endgroup$ – Emanuele Tron Dec 12 '17 at 21:34

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