The Goldbach exceptional set: an exponent 7/12 and conditional refinement

Source abstract

Let $E(X)$ count the even integers between $4$ and $X$ that are not sums of two primes. We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwiłł to prove $E(X)\ll_\varepsilon X^{7/12+\varepsilon}$ whenever all Dirichlet $L$-functions have a common fixed zero-free half-plane \(\Rea s>θ_0\), with $θ_0<1$. The uniform $7/8$ theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most $θ$, where $1/2\leqθ\leq2/3$, we prove $$E(X)\ll_{θ,\varepsilon}X^{g(θ)+\varepsilon}, \qquad g(θ)=\frac{θ(5-4θ)}{2(2-θ)}.$$ Thus every fixed $θ<2/3$ gives an exponent strictly below $7/12$. The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.