OPENAI SAYS A FEARED ZERO NEVER EXISTS — A MATHEMATICIAN CHECKS

Conflict of interest. The paper states that the large language model Claude, made by Anthropic, was used throughout its preparation: it produced a report on OpenAI’s manuscript and an expanded draft of the proof that forms the basis of the text, worked out the explicit constants, and helped check the proof. This article is also written by Claude. The underlying result is credited to OpenAI, a competitor of Anthropic. Neither fact makes the proof right or wrong; that is for independent mathematicians to judge.

Number theorists study how prime numbers are distributed with tools called Dirichlet L-functions, L(s, χ), each attached to a “character” χ with a “conductor” q. The places where these functions vanish are their zeros.

For real characters, one awkward possibility has never been ruled out: a single real zero β lying extremely close to 1. It is called a Landau–Siegel zero. Whether such zeros exist, the paper notes, is one of the central problems of analytic number theory, because they block uniform estimates for primes in arithmetic progressions and effective lower bounds for the “class numbers” of quadratic fields.

The known partial answers all leave something out. Siegel, in 1935, showed that such a zero cannot be too close to 1, but the method gives no way to compute how close. Page showed that, up to any given size, at most one real character can have such a zero. Later work by Goldfeld, and by Gross and Zagier, gave effective bounds on class numbers — but, the paper says, far weaker than what ruling out these zeros would give.

OpenAI’s claim

A manuscript by OpenAI, dated 1 October 2026, asserts that Landau–Siegel zeros do not exist: there is a fixed constant c > 0 such that (1 − β) log q ≥ c for every real zero, where q is the “conductor” of the character. According to William Banks of the University of Missouri, its proof is quite short, uses an interpolation determinant — a tool familiar from the theory of transcendental numbers — and needs only one classical inequality about L-functions. But it argues by contradiction and gives no numerical value for any constant.

A proof written out in full

Banks claims no originality for the ideas. The goal is to make the argument accessible and to confirm it, as a direct proof in which every step is written out. What is new are explicit numbers:

(1 − β) log q ≥ 0.003 for every conductor q ≥ 10¹⁰

Combined with a computer search by Lu, Zaman and Zhao for smaller conductors — which Banks reports not having verified — the bound would hold for every conductor from 3 upward.

How the trap closes

The proof plays one whole number against itself.

A biased prime count. If a real zero sat very close to 1, then χ(p) = −1 for almost all primes p. This is classical, and it is the only place where the zero enters.
A determinant. In a number system built from the square root of 2 and the square root attached to χ, Banks assembles a huge determinant Δ from powers of carefully chosen numbers. Its “norm” is a nonzero integer.
Too big and too small at once. A size bound says this integer cannot be very large. But at every prime p with χ(p) = −1, an algebraic rule called the Frobenius congruence makes Δ divisible by high powers of p. If almost all primes behave this way — as a near-1 zero would force — the integer would need more prime factors than its size allows.

The contradiction disappears only if the zero keeps a minimum distance from 1. In Banks’s words, the key trick is that the auxiliary number 2 raises the rank of the lattice to four while keeping only three variables.

Now, the checking

The stakes are high, and Banks says so: “In view of the importance of the statement, independent scrutiny is essential; this paper is written to make that scrutiny as easy as possible.” Banks writes of having personally verified every line and believes the argument to be sound. The result has not yet been reviewed by other mathematicians, and much of this write-up was prepared with the help of an AI system. The next step belongs to the number theory community.