On 6 October 2026 OpenAI released a large collection of mathematics manuscripts written by an internal model. One of them claims a statement number theorists have wanted for a century: a single zero-free half-plane for every Dirichlet \(L\)-function. None of it has been refereed yet.
This note asks a narrower question. If that claim is true, what happens to the binary Goldbach problem? In short, the best bound we can prove for the number of even numbers up to \(X\) that are not a sum of two primes would drop from about \(X^{0.7}\) to \(X^{3/5+\varepsilon}\). Goldbach’s conjecture itself would remain open.
1. The claim
The manuscript states that every Dirichlet \(L\)-function, the Riemann zeta function included, has no zero in the half-plane \(\operatorname{Re} s>7/8\), apart from the pole of \(\zeta\) at \(s=1\). It comes with a Lean formalization whose final statement uses Mathlib’s own definition of Dirichlet \(L\)-functions. A companion manuscript excludes Landau–Siegel zeros, which the half-plane already implies.
Classical zero-free regions shrink towards the line \(\operatorname{Re} s=1\) as the modulus and the height grow, and they leave room for one exceptional real zero. A fixed half-plane \(\operatorname{Re} s>\Theta\) with \(\Theta<1\) is called the quasi-Riemann hypothesis. It is much weaker than the generalized Riemann hypothesis, which puts every nontrivial zero on \(\operatorname{Re} s=1/2\), and much stronger than anything previously known.
The claim is unrefereed and machine-generated, and the release itself says that some unformalized results could have issues. At the time of writing I know of no independent check of this one. Everything below is conditional on it.
2. The Goldbach exceptional set
Write \(E(X)\) for the number of even \(n\le X\) that are not a sum of two primes. Goldbach’s conjecture says that every even \(n\ge4\) is such a sum, so the only exception would be \(n=2\). What can be proved is that other exceptions, if any, are rare.
Pintz proved \(E(X)\ll X^{0.72}\), and Zhao improved this to \(E(X)\ll X^{7/10}\). My computer-assisted refinement of Zhao’s method gives \(X^{23/33+\varepsilon}\approx X^{0.69697}\) (the paper), and S. Zhou has posted \(183/263\approx0.6958\) on Zenodo. Neither refinement is refereed.
All of these bounds share one architecture. The circle method splits the number of representations of \(n\) into a major-arc main term and a minor-arc remainder. Pintz proved an explicit formula for the major arcs: the main term is \(\mathfrak S(n)\,n\), plus correction terms coming from generalized exceptional zeros. These are zeros of \(L\)-functions of conductor up to a fixed power of \(X\), lying within \(H/\log X\) of the line \(\operatorname{Re} s=1\). For special \(n\) those corrections can cancel the main term. The work of Pintz, Zhao and the later refinements is about bounding how often that can happen.
3. What the 7/8 claim would give
If no zero has real part above \(7/8\), then as soon as \(\log X>8H\) every zero satisfies \(1-\beta\ge1/8>H/\log X\). There are no generalized exceptional zeros at all, and Pintz’s formula reduces to its main term \(\mathfrak S(n)\,n\). Pintz marks this case in his own proof: “If \(K=0\) we are ready.”
The rest is classical. Taking major arcs of level up to \(X^{2/5}\), Vaughan’s minor-arc estimate leaves at most \(O(X^{3/5+\varepsilon})\) even \(n\le X\) at which the minor arcs could cancel the main term. Hence
\[E(X)\ll_\varepsilon X^{3/5+\varepsilon}.\]
The implied constant would also be effective: the only ineffective step in Pintz’s theorem is the branch that handles a Siegel zero, and that branch no longer occurs. Any fixed half-plane \(\operatorname{Re} s>\Theta\) with \(\Theta<1\) gives the same exponent \(3/5\); the value \(7/8\) plays no further role.
4. What it would not give
Goldbach itself. The exponent \(3/5\) is a limit of the minor arcs, not of the zeros. Below it, the exceptional count is controlled by the fourth moment of the prime exponential sum over the minor arcs. A bound beyond Vaughan’s there would push the exponent towards \(5/9\); the generalized Riemann hypothesis gives \(1/2\).
The parity problem. With every zero-related obstacle removed, binary Goldbach for all large \(N\) reduces to a single one-sided inequality: for each large even \(N\) separately, the minor-arc part of the circle-method integral must not cancel more than a fixed fraction of the main term \(\mathfrak S(N)\,N\). That is exactly where the parity barrier of sieve theory sits, and no zero-free region reaches it.
The rest of the release. The same release claims the two-point Chowla conjecture, but only for fixed linear forms such as \(\lambda(n)\lambda(n+2)\). The Goldbach correlation \(\lambda(n)\lambda(N-n)\) has a shift that grows with \(N\), so it is not covered. The release’s parity-breaking theorems about primes need the number next to the prime to carry many small prime factors. In Goldbach that number is itself a prime.
5. Goldbach up to parity
Write \(\lambda(n)=(-1)^{\Omega(n)}\), where \(\Omega(n)\) counts the prime factors of \(n\) with multiplicity. Mangerel proved, assuming the generalized Riemann hypothesis, that every large even \(N\) can be written \(N=a+b\) with \(\lambda(a)=\lambda(b)=-1\): both summands have an odd number of prime factors. His paper notes that a much smaller zero-free region suffices, and the \(7/8\) half-plane contains it. So this “Goldbach up to parity” would hold, with an effective threshold.
Primes are the special case of exactly one prime factor. Getting from “an odd number” to “exactly one” is precisely the parity problem.
6. Twin primes
Pintz’s explicit formula also covers differences of primes. The same argument would show that all but \(O(X^{3/5+\varepsilon})\) even numbers \(m\le X/4\) are differences \(p-p'\) of primes up to \(X\). It says nothing about the single gap \(m=2\), so the twin prime conjecture is untouched.
7. Where this leaves things
If the \(7/8\) claim survives scrutiny, the exceptional-set problem stops being a delicate optimization over zeros near \(\operatorname{Re} s=1\) and becomes a clean question about the minor arcs, and my \(23/33\) bound becomes a historical step. Until the claim is checked, the unconditional bounds stand as they were. Either way, Goldbach’s conjecture remains open.
The most useful next step is independent verification. The Lean formalization can be built and checked by anyone, and an independent build would be the first real test.
References
- OpenAI, Mathematics manuscript collection, GitHub, 6 October 2026, in particular The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) > 7/8. github.com/openai/math.
- J. Pintz, A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach problem and the generalized twin prime problem, Acta Arith. 210 (2023), 53–94. doi:10.4064/aa220728-31-3.
- J. Pintz, A new explicit formula in the additive theory of primes with applications II. The exceptional set in Goldbach’s problem, arXiv:1804.09084. arxiv.org/abs/1804.09084.
- G. Zhao, The exceptional set of Goldbach problem and Linnik’s constant, arXiv:2511.05631. arxiv.org/abs/2511.05631.
- A. P. Mangerel, On Shusterman’s Goldbach-type problem for sign patterns of the Liouville function, arXiv:2412.17199. arxiv.org/abs/2412.17199.
- S. Zhou, Position-preserving zero-packet optimization for the binary Goldbach exceptional set, Zenodo, 2026. zenodo.org/records/22072882.
- L. Schiavone, A computer-assisted 23/33 + ε bound for the exceptional set in the binary Goldbach problem, 2026. lorenzoschiavone.com/writing/goldbach-exceptional-set-bound.