A note on the Cramér-Granville model

TL;DR

The paper demonstrates that a carefully constructed probabilistic subset $A$ of the integers exists with Bateman--Horn type prime-counts for polynomial systems, Goldbach representations, and a nontrivial lower bound on the prime density within $A$, all obtained via the Kim--Vu concentration inequality in a fixed-measure Cramér--Granville model. It combines analytic number theoretic conjectures with probabilistic methods to produce almost-sure asymptotics matching the conjectural predictions. The main contributions are establishing a unified random-model framework that encodes BH predictions, Goldbach-type decompositions, and prime-density improvements inside a single set $A$, with explicit error terms. This approach sheds light on how conjectural prime distributions might be realized inside structured subsets, with potential implications for extremal problems in prime theory.

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 \[ \#\{p\leq x \text{ prime} ~|~ p\in A\} \gg x(\log\log x)/(\log x)^2. \]

Theorems & Definitions (7)

• Theorem 1.1
• Remark
• Lemma 2.1: Kim--Vu kimvvu00
• Lemma 2.2
• proof
• Lemma 2.3
• proof