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arXiv · 2607.27282

The exceptional set of the Goldbach problem

Gautami Bhowmik·Lasse Grimmelt
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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.

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Gautami Bhowmik, Lasse Grimmelt. 2026-07-29. The exceptional set of the Goldbach problem. https://arxiv.org/abs/2607.27282

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