The exceptional set of the Goldbach problem

The problem concerns the exceptional set $\mathcal{E}(X)$ of even integers up to $X$ that are not representable as the sum of two primes. Eliminating this set would settle the binary Goldbach conjecture, but current work establishes bounds rather than its emptiness.

Known results

• Montgomery and Vaughan (1975) proved an effective power-saving bound $|\mathcal{E}(X)|\leq X^{1-\delta}$ for some effectively computable $\delta>0$.
• Pintz later made the exponent explicit, with $\delta=0.28$.
• Under the hypothesis that Dirichlet $L$-function zeros have real part at most $\Theta<3/4$, the 2026 paper records $|\mathcal{E}(X)|\ll_{\varepsilon}X^{2\Theta-\frac12+\varepsilon}$; this remains conditional.

2026 explicit major-arc formula

Bhowmik and Grimmelt add a fully explicit major-arc formula retaining contributions from Dirichlet $L$-function zeros. They also show conditionally that a sufficiently sparse Hardy–Littlewood estimate for $r_2(N)$ rules out sufficiently close exceptional real zeros, including Siegel zeros. Neither result proves that $\mathcal{E}(X)$ is empty, and the paper’s claims have not been independently verified here.

Current status (as of September 2026): Quantitative unconditional bounds and new conditional information are known, but whether $\mathcal{E}(X)$ is empty for every $X$ remains open.