Th\'eor\`eme d'Erd\H{o}s-Kac dans presque tous les petits intervalles

We show that the Erd\H{o}s-Kac theorem is valid in almost all intervals $\left[x,x+h\right]$ as soon as $h$ tends to infinity with $x$. We also show that for all $k$ near $\log\log x$, almost all interval $\left[x,x+\exp\left(\left(\log\log x\right)^{1/2+\varepsilon}\right)\right]$ contains the expected number of integers $n$ such that $\omega(n)=k$. These results are a consequence of the methods introduced by Matom\"aki and Radziwi\l\l\ to estimate sums of multiplicative functions over short intervals.