The exceptional set of Goldbach problem

TL;DR

The paper tackles the exceptional set in the binary Goldbach problem by proving $E(X)=O(X^{0.709})$ with an ineffective constant, improving the exponent bound from previous work. Its core strategy refines Pintz's zero-analysis of Dirichlet $L$-functions to a two-dimensional Linnik-type framework focused on a single modulus, combined with a dichotomy that eliminates many zeros. A key technical advance is a Laplace-transform approach to bound weighted sums of zeros, with a sharp treatment when the zero-conductor set has bounded complexity, yielding explicit bounds across six case-distinct regimes. Together these elements bound the zero-sum terms $\sum_i (\sum_{\lambda\in \mathcal L_i} e^{-\delta^{-1}\lambda})^2$ below 1, hence establishing $E(X)=O(X^{0.709})$ and advancing unconditional progress toward the binary Goldbach problem.

Abstract

Let $E(X)$ be the number of even integers below $X$ which are not a sum of two primes. We prove that $E(X)=O( X^{0.709})$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof