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 $[x,x+h]$ as soon as $h$ tends to infinity with $x$. We also show that for all $k$ near $\log \log x$, almost all interval $[x,x+\exp \left({(\log \log x)}^{1\mathrm{/}2+\epsilon }\right)]$ 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.