For $n\in \mathbb{N}$ with prime decomposition $n=p\_1^{r\_1}\cdots p\_k^{r\_k},p\_i\neq p\_j$, let $A=\{p\_1,\cdots,p\_k\}$; then the following holds:
\begin{equation}
|\{q\in \mathbb{N},q<Q: \text{all prime factors of } q\,\in A\}|<Ce^{c\frac{\log Q}{\log \log Q}}d(n,Q)
\end{equation}
where $d(n,Q):=|\{m\in \mathbb{N},m<Q: m\mid n\}|$.


This lemma appears below (3.46) in Bourgain's article "Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations".



> 
> I'm not familiar with number theory; could you please explain how to
> deduce this conclusion and give some references on number theory?
> 
> 
> 



Estimates about prime numbers: a lemma in Bourgain's article
Let $q<Q$ be such that all its prime divisors are in $A$. We can write $q=q\_0\times r\_1^{n\_1}\cdots r\_s^{n\_s}$, where all prime factors of $q\_0$ are $<\log Q$ and $\log Q\leq r\_1<\cdots<r\_s\in A$. Clearly, $q$ is uniquely determined once we have fixed a) $q\_0$, b) the set $\{r\_1,\ldots,r\_s\}$, and c) the integer vector $(n\_1,\ldots,n\_s)$.


The number of possible $q\_0$ is at most the number of $q<Q$ which have all prime factors $<\log Q$. This is a well-studied quantity in analytic number theory (counting 'smooth' or 'friable' numbers), and is $\leq \exp(O(\log Q/\log\log Q))$, which can be shown via elementary methods, see any textbook on analytic number theory.


There are clearly at most $d(n,Q)$ choices for the set $\{r\_1,\ldots,r\_k\}$.


Finally, to estimate the number of choices for (c), note that
$$\sum n\_i\log r\_i < \log Q,$$
and $\log r\_i\gg \log \log Q$ by construction, hence it suffices to bound the number of positive integers $n\_1,\ldots,n\_s\geq 1$ (where $s$ may vary) such that $\sum n\_i \ll \log Q/\log\log Q$. This is at most $\exp(O(\log Q/\log\log Q))$, using the elementary fact (simple combinatorics, 'stars and bars') that the the number of sequences of positive integers $n\_i$ with $\sum n\_i \leq M$ is exactly $2^M$.


Thus combining these three estimates, the number of possible $q$ is at most


$$ (\exp(O(\log Q/\log\log Q)))^2\times d(n,Q) \ll \exp(O(\log Q/\log\log Q))d(n,Q).$$


