Reducing the number of prime factors of long $\kappa$-tuples

Abstract

We prove that there are infinitely many integers $n$ such that the total number of prime factors of $(n+h_1)(n+h_2)...(n+h_{\kappa })$ is at most $(1/2)\kappa \log \kappa +O(\kappa )$, provided $\kappa$ is sufficiently large.

arXiv comment15 pages, submitted to London Mathematical Society