Can anyone give an estimate (upper bound or lower bound) for the number of divisors $d\mid P\_r$ such that $\frac{\sqrt{P\_r}}{2}< d < \sqrt{P\_r}$, where $P\_r$ is the product of the $r$ smallest primes?



Estimate about primes
Let $x$ be such that $\pi(x) \sim \frac{x}{\log x} \sim r$, so that $P\_r$ is about $e^x$. Let $\Psi(x,y)$ be the de Bruijn, function, the number of [$y$-smooth](https://en.wikipedia.org/wiki/Smooth_number) numbers less than or equal to $x$. So a crude upper bound for the quantity in question is about


$$
\Psi(e^{x/2},x) - \Psi(e^{x/2}/2,x).
$$


Out of these numbers (call the set of these numbers $X$) we want to get the squarefree ones. The proportion of squarefree numbers in all the integers is $1/\zeta(2)$ so it is tempting to use $|X|/\zeta(2)$ or $|X|\prod\_{p\leq x}\left(1-1/p^2\right)$ as a crude first approximation, though I'm not sure how far off it is. The thing is that one is dealing here with only $x$-smooth numbers. Perhaps for each $p \leq z \ll x$ for suitable $z$ one can estimate the size of the subset of $X$ that is in $p^2\mathbb{Z}$, then use a simple inclusion-exclusion sieve to estimate the size of the subset of $X$ of numbers that, if they are divisible by prime squares $p^2$, must have $z \leq p\leq x$. Or, use more powerful sieve methods. (Guessing that sieving for squarefree numbers should be very similar to sieving for primes.)


Good references:


A. Granville, Smooth numbers: computational number theory and beyond, Algorithmic Number Theory, MSRI Publications, Volume 44, 2008.


T. Tao's blog, 254B, Notes 7: Sieving and expanders.


