Small gaps between Goldbach primes

TL;DR

This paper compares two methods for studying small gaps between Goldbach primes, showing that the Bombieri-Davenport method can bound the smallest gap relative to the average, while the Maynard-Tao method confirms the existence of bounded gaps for most even integers.

Contribution

It demonstrates the effectiveness of the Bombieri-Davenport method in bounding small gaps between Goldbach primes and explores the limitations and potential of the Maynard-Tao method in this context.

Findings

Small gaps are at most 0.765 times the average gap for almost all even integers.

The Maynard-Tao method confirms the existence of bounded gaps with bounded error for most even integers.

The straightforward application of Maynard-Tao is insufficient to improve the gap bound.

Abstract

We study small gaps between Goldbach primes $\mathbb{P} \cap (N-\mathbb{P})$ using the Bombieri-Davenport method and the Maynard-Tao method, and compare the two. We show that for almost all even integers $N$, the smallest gap in $\mathbb{P} \cap (N-\mathbb{P})$ is at most $0.765\ldots$ times the average gap, using the Bombieri-Davenport method. This improves a recent result of Tsuda. We also demonstrate that a straightforward application of the Maynard-Tao method is insufficient to improve this bound. However, it allows us to establish the existence of bounded gaps between Goldbach primes with bounded error for almost all even integers $N$.