Square-free numbers in an interval Is there an explicit estimate in the literature bounding from above the number of square-free numbers in a short interval $x<n\leq x y$? I can easily do this by means of the Selberg sieve, but I do not want to reinvent the wheel.
 A: Let me just show how to derive a simple bound that has been mentioned in the comments. We are trying to bound the estimate the number $Q(x,x+u)$ of squarefree integers in $(x,x+u]$.
We can now apply Selberg's sieve (as in Iwaniec-Kowalski (say), Thm. 6.4,
with $g(d) = 1/d^2$, $X = u$, $r_d = 2$ and
$|\{x<m\leq x+u: \max \{d: d^2|m\} = n\}|$,
$D$ to be set later).
We obtain
$$Q(x,x+u)\leq
 \frac{u}{H} + 2 \sum_{d\leq D} |\lambda_d|,
 $$
where
$\lambda_d = \sum_{[d_1,d_2]=d} \rho_{d_1} \rho_{d_2}$ and
$|\rho_d|\leq 1$ for all $d$, $\rho_d=0$ for $d>\sqrt{D}$ or $d$ not
square-free, and
$$\begin{aligned} H &= \sum_{d\leq \sqrt{D}} \frac{\mu^2(d)}{\prod_{p|d} p^2 (1-1/p^2)} =
 \sum_{d\leq \sqrt{D}} \mu^2(d) \sum_{m: \textrm{rad}(m)=d} \frac{1}{m^2}\\&\geq
 \sum_{d\leq \sqrt{D}} \frac{1}{d^2} \geq \zeta(2) - \frac{1}{\sqrt{D}} -
 \frac{1}{D}\end{aligned}$$
with $\textrm{rad}(m) = \prod_{p|m} p$. It does not
take much work (just some casework for $D<4$) to show that in fact $H\geq \zeta(2)-1/D$ for $D\geq 1$. Clearly
$\sum_{d\leq D}
 |\lambda_d| \leq \left(\sum_{d\leq \sqrt{D}} \rho_d\right)^2
 \leq D.$ Since $1/H\leq (\zeta(2) (1-1/(\zeta(2) \sqrt{D})))^{-1}$, it follows that
$$\begin{aligned}Q(x,x+u)&\leq \frac{u}{\zeta(2)} + 2 D + \frac{u}{\zeta(2)^2 \left(1- \frac{1}{\zeta(2) \sqrt{D}}\right)\sqrt{D} }\\  &< \frac{u}{\zeta(2)} + 2 D + \frac{u}{\sqrt{D}}.\end{aligned}$$
We set $D = (u/4)^{2/3}$, and obtain
$$Q(x,x+u) < \frac{u}{\zeta(2)} + \frac{3}{2^{1/3}} u^{2/3}.$$
(Please feel free to point how this proof can be made simpler or shorter.)
A: Also: see Cohen-Dress, "Estimations numériques du reste de la fonction sommatoire relative aux entiers sans facteurs carrés", 1988 (https://www.imo.universite-paris-saclay.fr/~biblio/numerisation/docs/Colloque-theorie-analytique-nombres_J.Coquet_1985/pdf/Colloque-theorie-analytique-nombres_J.Coquet_1985.pdf), which gives several estimates, none of them strictly better or worse than the one in my answer (each one seems to be best in some range).
