The exceptional set of the Goldbach problem
arXiv:2607.27282
The paper surveys known results on the size of the exceptional set for representing integers as sums of at most two primes, reviews the Hardy‑Littlewood circle method and later power‑saving advances, and introduces a new explicit formula for the major arcs together with a result on the non‑existence of exceptional zeros under a sparse Hardy‑Littlewood conjecture.
Abstract
We study the estimates for the number of exceptions to the representation of integers as the sum of at most two prime numbers. Most of this article is a survey that gives an overview of existing results. We begin with the legendary Hardy-Littlewood circle method and show how it paved the way to a power saving by Montgomery-Vaughan in 1975 and Pintz in 2018. We conclude with a new result that is a fully explicit formula for the major arcs. Another new observation is the non-existence of exceptional zeros under a sparse version of the Hardy-Littlewood conjecture. The survey part of this article aims to be accessible to an audience that has not encountered these techniques before.
Typographic errors corrected. To appear in Analysis Mathematica