On the Density of Weak Polignac Numbers

TL;DR

This paper proves that the set of weak Polignac numbers, which are differences between primes occurring infinitely often, has positive density, improving existing bounds under a strong hypothesis.

Contribution

Provides a new unconditional proof of positive density for weak Polignac numbers and improves bounds assuming a strong hypothesis.

Findings

Weak Polignac numbers have positive density.

Improved bounds on the distribution of prime gaps.

New proof techniques under strong hypotheses.

Abstract

Let $k$ be an integer which is the difference between prime numbers infinitely often. It is known that there are infinitely many such $k$ and, in this paper, we give a new unconditional proof that these $k$ have positive density and improve on current bounds, assuming a strong hypothesis.