On long $\kappa$-tuples with few prime factors

Olivier Ramaré (1, 2)

1 CNRS - Centre National de la Recherche Scientifique
2 LPP - Laboratoire Paul Painlevé - UMR 8524

We prove that there are infinitely many integers n such that the total number of prime factors of $(n + h_1) \cdots (n + h_\kappa)$ is exactly $(1 + o(1))\kappa\log\kappa$. Our result even ensures us that these prime factors are fairly evenly distributed among every factors $n + h_i$.

Domaines: Théorie des nombres [math.NT]

Olivier Ramaré. On long $\kappa$-tuples with few prime factors. Proceedings of the London Mathematical Society, 2012, 104 (1), pp.158-196. ⟨10.1112/plms/pdr026⟩. ⟨hal-02572799⟩