A note on the Cram\'er-Granville model

TL;DR

This paper demonstrates the existence of a special set of integers that aligns with predictions from major conjectures like Bateman-Horn and Goldbach, and also has a significant density of primes.

Contribution

It constructs a set satisfying multiple major conjectural estimates and provides lower bounds on its prime elements, advancing understanding of prime distributions.

Findings

Set A satisfies Bateman-Horn and Goldbach conjecture estimates.

Number of primes in A grows faster than x(log log x)/(log x)^2.

Provides a new example related to prime distribution conjectures.

Abstract

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