Théorème d'Erdős-Kac dans un régime de grande déviation pour les translatés d'entiers ayant $k$ facteurs premiers
Let $x\geqslant 3$, for $1\leqslant n \leqslant x$ an integer, let $ω(n)$ be its number of distinct prime factors. We show that, among the values $n\leqslant x$ with $ω(n)=k$ where $1\leqslant k \ll \log_2 x$, $ω(n-1)$ satisfies an Erdős-Kac type theorem around $2\log_2 x$, so in large deviation regime, when weighted by $2^{ω(n-1)}$. 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.
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