Théorème d'Erdős-Kac dans un régime de grande déviation pour les translatés d'entiers ayant k facteurs premiers

2024/10/15 by Garçonnet, Olivier
#11N05 #11N13 #11N25 (Primary) #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.11616

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 ≪ log₂ x, ω(n-1) satisfies an Erdős-Kac type theorem around 2log₂ 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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