OpenAI's quasi-Riemann claim: how the boundary would move

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An AI system could make a consequential contribution to number theory while leaving the Riemann hypothesis open. A lasting restriction on where zeros can occur would give later researchers something new to use. We think that is the significance worth investigating in OpenAI's quasi-Riemann claim. It asks whether a machine-attributed argument could change the work a field can do, well short of resolving its most famous question.

Riemann's 1859 paper connected the distribution of primes to the zeros of a complex function. One hundred and sixty-seven years later, the full Riemann hypothesis remains open. Mathematicians have nevertheless learned a great deal by restricting where zeros can occur, even when they cannot put every zero on the conjectured line.

OpenAI is claiming an empty region that stays empty, however far up an entire family of L-functions we look. Its 6 October release attributes the work to an internal model. The stronger manuscript proposes a fixed boundary at seven-eighths, where familiar zero-free regions become narrower with increasing height. It does not claim to settle the full Riemann hypothesis.

We followed the historical approaches and the two released arguments to understand how OpenAI tries to obtain that permanence. The companion makes the role of arithmetic cancellation visible. The stronger paper compares two representations of the same sum, then tries to control the terms that could conceal a zero. Those estimates are where the case for a new contribution has to be made, before we can consider what other mathematics it might enable.

Riemann's route from primes to zeros

The historical starting point is the uneven spacing of primes. We know they thin out, but an average trend leaves room for fluctuations. How tightly can mathematics control those fluctuations?

In his six-page paper of 1859, Bernhard Riemann connected prime counting to the zeros of the zeta function. A zero here is a complex input where the function vanishes, rather than a missing prime on the number line. The unexpected move was to study those inputs in the complex plane to understand ordinary whole numbers.

Jacques Hadamard and Charles-Jean de la Vallée Poussin independently proved the prime number theorem in 1896. Their work established the asymptotic law: the number of primes up to x behaves like x/log x. The full Riemann hypothesis would give much tighter control over departures from that average. It makes the locations of zeros consequential, rather than merely a pattern to plot.

We think that is the right setting for OpenAI's claim. The proposed strip is weaker than Riemann's critical line, but it would impose a lasting restriction on the zeros across a much larger family of functions. The question is how much prime-counting control that restriction can earn.

Seven-eighths isn't a progress percentage

Write a complex input as $s=\sigma+it$. The real part $\sigma$ is its horizontal position; $t$ gives its height. The Riemann hypothesis places all nontrivial zeta zeros on $\sigma=\tfrac12$.

OpenAI's 30 September manuscript claims there are no zeros for $\sigma>\tfrac78$ for every Dirichlet L-function and every finite-order Hecke L-function over $\mathbb{Q}(\sqrt{-3})$. The principal pole at $s=1$ is allowed; the boundary itself isn't included in the exclusion.

Its 5 October companion claims $\sigma>\tfrac{11}{12}$ using a different argument. Moving the boundary left excludes more possible zeros. We used three hypothetical points to make that difference visible:

Real part of a hypothetical zero	Excluded by $\sigma>\tfrac78$?	Excluded by $\sigma>\tfrac{11}{12}$?
0.95	Yes	Yes
0.90	Yes	No
0.75	No	No

These aren't computed zeros. They show why seven-eighths mustn't be sold as a fraction of the Riemann hypothesis solved. A zeta zero at 0.75 would violate the full hypothesis while fitting both claimed bounds.

We think the mathematical case is stronger when it keeps that distinction. A fixed excluded region has value without pretending to deliver the critical line.

The boundary would stop retreating

One human approach bounds exponential sums: sums of oscillating complex terms that help control the size of zeta near the line with real part 1. Chiara Bellotti's 2023 work improves such estimates, then combines them with nonnegative trigonometric polynomials to exclude zeros. The resulting control includes logarithmic losses depending on height. Its explicit zero-free boundary has the form

$$\sigma>1-\frac{1}{54.004\,(\log|t|)^{2/3}(\log\log|t|)^{1/3}}.$$

Here $|t|$ measures height, and this expression applies for $|t|\ge 3$. As height grows, the subtracted quantity tends to zero. The boundary approaches 1.

That comparison made the proposed advance clearer to us. A permanent boundary at 7/8 would keep ruling out the same horizontal region at arbitrarily large heights. Across all Dirichlet characters, it would also survive increasing conductors, the arithmetic parameters indexing the family.

A second route studies how often Dirichlet polynomials—finite versions of the relevant series—can become large. Larry Guth and James Maynard's 2024 work improves those large-value bounds and obtains stronger estimates for the number of zeros in specified regions. A bound on the number can still permit individual zeros. OpenAI's fixed-region claim needs to exclude every one of them.

Nor does uniformity require every constant to be identical. Multiplying constants and starting thresholds can depend on the character while the exponent stays fixed. The exponent is what supplies the shared boundary here.

These are distinct obstacles. The first comparison leaves a margin that shrinks with height. The second controls a count without forcing it to be zero. The released argument has to establish a fixed positive margin while retaining the arithmetic uniformity its theorem asserts.

How OpenAI tries to force the exclusion

The companion gives an approachable route into the proof idea through the Möbius function. It assigns zero to integers with a repeated prime factor, and alternates signs according to the number of distinct prime factors. For $\sigma>1$:

$$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}.$$

Let $M(x)=\sum_{n\le x}\mu(n)$ be the sum up to $x$. Partial summation rewrites the reciprocal as

$$\frac{1}{\zeta(s)}=s\int_1^{\infty}M(x)\,x^{-s-1}\,\mathrm{d}x.$$

Suppose we could prove $|M(x)|\le Cx^\theta$ for a fixed $\theta<1$. Then the integral converges for $\sigma>\theta$ and supplies a holomorphic continuation of the reciprocal there. A zeta zero would make the reciprocal have a pole, contradicting that continuation.

The difficulty is the cancellation. Taking absolute values only gives a bound of order x. To save a fixed power, the positive and negative contributions must keep cancelling enough at arbitrarily large scales.

The companion extends this cancellation idea to sums weighted by arithmetic characters. A character assigns structured weights to numbers or ideals; an L-function packages those weights into a series related to zeta. The proposed estimates must retain cancellation after the weights are introduced.

The stronger 7/8 manuscript uses a different comparison, described in its proof overview and Sections 17–20. It constructs a smoothed cubic-theta sum and writes two exact representations of it. Reflection supplies a direct upper bound. Poisson summation, which converts a lattice sum into frequency contributions, exposes a principal contribution containing the reciprocal of the target L-function.

The paper therefore tries to isolate the dangerous reciprocal by comparing two descriptions of the same object. It inherits theta-function machinery from Kubota and Patterson, as well as later unconditional cusp expansions and large-sieve estimates. Its proposed addition is a compatible comparison strong enough to retain the principal contribution while controlling all the others. Poisson summation itself is an established human tool.

This creates a route to a contradiction. If the remaining frequency contributions are small enough, the two representations force the reciprocal to continue regularly to the left of the supposed rightmost-zero boundary. A zero would create a singularity there. The proof must obtain a positive power saving common to the comparison, rather than another margin that disappears at high scales.

The first stage of the stronger manuscript targets 11/12 with symmetric scales. The improvement to 7/8 changes the averaging ranges and inserts selected prime factors together with compensating terms. Second- and fourth-moment estimates bound averages of powers of the troublesome quantities, giving stronger control than a typical-size estimate alone. The paper must make those separate estimates work together without losing the common positive saving.

The important transition is from those aggregate estimates to a fixed zero-free boundary. The saving has to survive interacting parameters, exceptional contributions and the limiting argument. Proposition 20.3 separates shared parameter choices from character-dependent ones. We haven't certified those analytic steps. They identify where the universal promise has to be established, rather than merely illustrated by favourable calculations.

What would the promise buy?

Zeros matter because explicit formulas connect them to prime distribution. Bombieri's official problem description explains that relationship. Excluding a fixed right-hand region would constrain contributions from zeros close to 1, even while the stronger critical-line ambition remains open.

The companion claims an effective absolute error of order $x^{11/12}\log x$ for prime counts in reduced arithmetic progressions, uniformly over moduli $q\le x$. That is still a manuscript claim. An absolute bound also needn't make a good relative estimate when the expected count is small: the allowed error can dwarf it.

We see the potential contribution as a stronger arithmetic toolkit. The useful question is what this permanent restriction enables mathematicians to prove next.

The release offers a Lean entry for the zeta 7/8 statement. We inspected its declaration and comparator configuration, but haven't checked the full import chain or run the kernel. That provides another route to verification, not a verification we completed.

Progress without the final conjecture

An established fixed zero-free region would become part of the conditions under which number theorists can work. As the earlier prime-counting discussion shows, its value would depend on the estimates it actually supplies, including their dependence on the character. The field could gain a useful result even while the full Riemann hypothesis remained unresolved.

We would judge the proposed advance by whether researchers can use its argument to reach results they could not previously establish. A famous conjecture is an easily recognised target. A stronger estimate can have a less obvious public story while changing the assumptions available to many subsequent arguments. That is one reason to resist treating the list of celebrated problems solved as a complete account of AI's contribution to research.

The discovery story still needs qualification. OpenAI's disclosure names zeta zero-free work as an exception to its standard generation procedure and says the 11/12 write-up was human edited for readability. We can reconstruct the proposed strategy, but the public record does not establish the full division of labour. The mathematical argument also inherits substantial human work on cancellation, theta functions and analytic estimates.

For research communities, the larger possibility is an additional source of the intermediate advances on which further discoveries depend. That could alter how people choose problems and collaborate. A group might be able to investigate a direction that had been blocked by an estimate nobody could obtain. The useful achievement would be the argument that removes that obstacle, rather than the mere production of another manuscript.

Scientists outside mathematics can ask a related question about research agents. Can a proposed contribution become a sound basis for the next investigation? In an empirical field, the answer also depends on measurement and experiment. Success in number theory does not answer those questions for biology or physics, but it gives us a reason to examine the inventive capabilities involved instead of assuming them to be exclusively human.

A new zero-free region could matter through the work it enables long after this announcement. We would regard that as a serious contribution from a research system, even while the original Riemann question remained open. For number theorists, the prospect would be a larger range of questions they can seriously attempt. That would be a substantial consequence of a partial result.

Source note: we read selected theorem statements and proof overviews at OpenAI's release snapshot adc7f1241b42e322a6451854ab7e4b4c146bf78a, compared the primary references above and constructed the hypothetical-point table. Sources checked 8 October 2026. We haven't certified the analytic proofs or completed Lean verification.