Facteurs premiers de sommes d’entiers

Abstract:

Let $\Omega (n)$ denote the total number of prime factors of the positive integer $n$, and set for real $x$, $\alpha$, and integer $k \geq 1$, \[ \pi _k(x) = \sum \limits _{\substack {n \leqslant x \\ \Omega (n) = k}} 1, \quad \pi _k(x, \alpha ) = \sum \limits _{\substack {n \leqslant x \\ \Omega (n) = k}} \mathbf {e}(\alpha n), \quad E(x,\alpha ) = x^{-1}\sum \limits _{n \leqslant x} \mathbf {e}(\alpha n),\] where ${\mathbf {e}}(t): = \exp (2\pi it)$. We establish a best possible "independence" result of the type \[ {\pi _k}(x,\alpha )/{\pi _k}(x) = E(x,\alpha ) + O({\delta _k}(x))\] which is valid uniformly in $x,k,\alpha$, and where the error ${\delta _k}(x)$ tends to 0 as $x \to + \infty$, if, and only if, $k \sim \operatorname {log} \operatorname {log} x$. As an application we prove a recent conjecture of Erdös, Maier, and Sárközy concerning the remainder in their Erdös-Kac theorem for sum-sets.