Conditional bounds for the Goldbach exceptional set

Let E(X) count Goldbach exceptions up to X. How small an exponent can bound E(X) assuming one common fixed zero-free half-plane Re(s)>θ₀, θ₀<1, for all Dirichlet L-functions? A stronger zero-location hypothesis Re(ρ)≤θ∈[1/2,2/3] asks for a bound depending explicitly on θ.

Why it matters and what counts as progress

Why it matters: Conditional exceptional-set exponents quantify what a uniform Dirichlet-zero hypothesis would imply about Goldbach failures.

What counts as progress: Lower the conditional exceptional-set exponent, including the targets 7/12+ε and θ(5−4θ)/(2(2−θ))+ε, without weakening or conflating the analytic hypotheses.

How the problem developed

2018 · Partial result [1]
János Pintz Derived an exceptional-set upper bound of order X^0.72 from an explicit formula for additive prime problems.
11 Oct 2026 · Conditional bound [2]
János Pintz and Ingo Althöfer Derived E(X) ≪ X^(7/12+ε) under a common fixed zero-free half-plane for Dirichlet L-functions; a stronger zero-location hypothesis gives a better exponent.

AI contribution

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

János Pintz, Ingo Althöfer

The explicit conditional exceptional-set exponent g(theta)=theta(5-4theta)/(2(2-theta)), for the stated common Dirichlet-zero hypothesis theta in [1/2,2/3]. The separate 7/12 argument and earlier numerical chain are not the AI contribution.

AI involvementAI-centered

ImportanceNotable

Technical verificationPartial

Human understandingFew