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

TL;DR

This paper extends the Erdős-Kac theorem to almost all short intervals growing with x, showing that the normal distribution of the number of prime factors holds in these intervals, using advanced methods for multiplicative functions.

Contribution

It demonstrates the Erdős-Kac theorem's validity in almost all short intervals as they grow with x, employing novel techniques from Matomäki and Radziwiłł.

Findings

Erdős-Kac theorem holds in almost all intervals as h tends to infinity.

Almost all intervals of specified length contain the expected number of integers with a given number of prime factors.

Methods from Matomäki and Radziwiłł are effective for estimating sums of multiplicative functions in short intervals.

Abstract

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.