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

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.

9 pages. Final version. Some minor changes, Analytic Number Theory - In Honor of Helmut Maier's 60th Birthday, Springer, 2015

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