Lois locales de la fonction $\omega$ dans presque tous les petits intervalles

TL;DR

This paper studies the distribution of integers with a fixed number of prime factors within small intervals, providing optimal and near-optimal results for various ranges of prime factor counts, extending recent advances in the field.

Contribution

It offers new asymptotic estimates for the count of integers with a given number of prime divisors in very short intervals, improving understanding of their local distribution.

Findings

Optimal results for $k hickapprox \log_2 x$

Near-optimal results for $5 \leq k hickapprox \log_2 x$

Method applies to $y$-friable integers in small intervals

Abstract

For $k\geq 1$ an integer and $x\geq 1$ a real number, let $\pi_k(x)$ be the number of integers smaller than $x$ having exactly $k$ distinct prime divisors. Building on recent work of Matom\"aki and Radziwi\l\l, we investigate the asymptotic behavior of $\pi_k(x+h)-\pi_k(x)$ for almost all $x$, when $h$ is very small. We obtain optimal results for $k\asymp\log_2 x$ and close to optimal results for $5\leq k\leq\log_2 x$. Our method also applies to $y$-friable integers in almost all intervals $[x,x+h]$ when $\frac{\log x}{\log y}\leq (\log x)^{1/6-\varepsilon}$.