Théorème d’Erdős–Kac dans presque tous les petits intervalles
Volume 182 / 2018
Acta Arithmetica 182 (2018), 101-116
MSC: Primary 11N25.
DOI: 10.4064/aa8480-11-2017
Published online: 11 January 2018
Abstract
We show that the Erdős–Kac theorem is valid in almost all intervals $[x,x+h]$ as $h$ tends to infinity with $x$. We also show that for all $k$ near $\log\log x$, almost all intervals $[x,x+\exp((\log\log x)^{1/2+\varepsilon})]$ contain the expected number of integers $n$ such that $\omega(n)=k$. These results are a consequence of the methods introduced by Matomäki and Radziwiłł to estimate sums of multiplicative functions over short intervals.
