# Th\'eor\`eme d'Erd\H{o}s-Kac pour les translat\'es d'entiers ayant $k$ facteurs premiers

## TL;DR

This paper extends the Erdős-Kac theorem to the number of distinct prime factors of n-1 for integers n with a fixed number of prime factors, under certain growth conditions, generalizing previous results.

## Contribution

It establishes an Erdős-Kac type theorem for \\omega(n-1) conditioned on \\omega(n)=k, for k up to a constant times log log x, extending Halberstam's work.

## Key findings

- entity of a normal distribution for \\omega(n-1) under the given conditions
- Extension of Erd\
- Theorem to a new class of integers with fixed prime factors

## Abstract

Let $x\geqslant 3$. For $1\leqslant n\leqslant x$ an integer, let $\omega(n)$ be its number of distinct prime factors. We show that $\omega(n-1)$ satisfies an Erd\H{o}s-Kac type theorem whenever $\omega(n)=k$ where $1\leqslant k\ll\log\log x$, thus extending a result of Halberstam.