A note on the Cramér-Granville model

Abstract

We show the existence of a set $A\subseteq {\mathbb{Z}}_{\ge 2}$ satisfying the estimates of the Bateman--Horn conjecture, Goldbach's conjecture, and also \[ \#\{p\leq x \text{ prime} ~|~ p\in A\} \gg x(\log\log x)/(\log x)^2. \]

7 pages