Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-Tao

TL;DR

This paper explores the implications of the Zhang-Maynard-Tao theorem for the density of prime k-tuples, establishing limits on what can be inferred about prime constellations based solely on this theorem.

Contribution

It determines the maximum possible densities of prime k-tuples that can be deduced from the Zhang-Maynard-Tao theorem, extending understanding of prime distributions.

Findings

Identifies the proportion of integers with known infinitely many prime pairs.

Generalizes the results to k-tuples of integers.

Establishes limits on what the Zhang-Maynard-Tao theorem can imply about prime densities.

Abstract

We determine for what proportion of integers $h$ one now knows that there are infinitely many prime pairs $p,\ p+h$ as a consequence of the Zhang-Maynard-Tao theorem. We consider the natural generalization of this to $k$-tuples of integers, and we determine the limit of what can be deduced assuming only the Zhang-Maynard-Tao theorem.