Th\'eor\`eme d'Erd\H{o}s-Kac dans un r\'egime de grande d\'eviation pour les translat\'es d'entiers ayant $k$ facteurs premiers

TL;DR

This paper proves an Erdős-Kac type theorem for the number of prime factors of integers shifted by one, in a large deviation regime, with weighted counts, refining previous results with sharper error bounds.

Contribution

It establishes a new Erdős-Kac theorem for shifted integers with a large deviation analysis, improving previous bounds and incorporating recent advances in divisor problems.

Findings

Erdős-Kac law holds for shifted integers with large deviations

Weighted counts by 2^{ω(n-1)} are analyzed

Quantitative error bounds are achieved

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, among the values $n\leqslant x$ with $\omega(n)=k$ where $1\leqslant k \ll \log_2 x$, $\omega(n-1)$ satisfies an Erd\H{o}s-Kac type theorem around $2\log_2 x$, so in large deviation regime, when weighted by $2^{\omega(n-1)}$. This sharpens a result of Gorodetsky and Grimmelt with a quantitative and quasi-optimal error term. The proof of the main theorem is based on the characteristic function method and uses recent progress on Titchmarsh's divisor problem.