Small gaps between Goldbach primes
Mizuki Akeno
TL;DR
The paper investigates how small gaps between Goldbach primes can be forced for almost all even integers N by comparing two leading analytic frameworks: the Bombieri–Davenport method and the GPY/Maynard–Tao approach. It shows that an explicit BD-driven bound of 0.76542… times the average gap is achievable for almost all N, aided by refined level-of-distribution results and a detailed analysis of correlations and singular series. Conversely, a straightforward Maynard–Tao adaptation cannot beat this BD bound, though it does establish the existence of bounded gaps with a bounded error for almost all N, with a concrete bound H ≤ 56250000. Overall, the work clarifies the respective capacities and limitations of these methods in the Goldbach-prime gap setting and provides explicit quantitative bounds and constructions that advance our understanding of Goldbach-type questions.
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 $N$.