Théorème d'Erdős-Kac dans un régime de grande déviation pour les translatés d'entiers ayant $k$ facteurs premiers
Olivier Garçonnet
TL;DR
The paper studies the Erdős–Kac type distribution for the shifted function ω(n-1) when n ranges over integers with ω(n) = k, with a fixed small- to moderate-range k (k ≤ R log_2 x), and under the weight 2^{ω(n-1)}. It develops a generating-series framework via F_k(z) and a multiplicative auxiliary g_z, handles large primes through Rankin truncation, and employs Fouvry–Tenenbaum theory alongside Berry–Esseen to obtain a precise Gaussian limit with a quantitative error term in the large-deviation regime, refining previous results of Gorodetsky–Grimmelt and Gou20 and extending to a fixed k. The main results include an explicit asymptotic for S_k(x) and a uniform local limit S_k(x,y) = S_k(x) Φ(y) + O((log_3 x)/√(log_2 x)) for 1 ≤ k ≤ R log_2 x, plus a corollary describing the distribution of small prime factors of n-1. The methods bridge Titchmarsh’s divisor problems with modern analytic techniques (generating functions, Selberg-type contour arguments, and Berry–Esseen), contributing sharp error terms and extending the Erdős–Kac philosophy to shifted multiplicative structures.
Abstract
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.