Mathematicians Checked the AI's Riemann Breakthrough

Main Ideas

The Quasi-Riemann Hypothesis

OpenAI announced a partial result related to the Riemann hypothesis. The full hypothesis states that every nontrivial zero of the zeta function lies on the line with real part (1/2). The AI’s result, called the quasi-Riemann hypothesis, establishes a weaker zero-free region: no zero lies to the right of the line with real part (7/8).

The zeta function’s zeros are closely connected to the distribution of prime numbers. Although the (7/8) bound is far from proving the full hypothesis, mathematician Maksym Radziwiłł said that many consequences of the full hypothesis already follow from the quasi-hypothesis.

Two Versions of the Proof

• The first version, 199 pages long, was checked line by line using the Lean proof assistant.
• A later, 49-page version proves a weaker bound of (11/12) and was not computer-checked. Mathematicians have focused on it because it is shorter and easier to study.
• According to the video, the shorter paper was the only one among 722 OpenAI releases edited by a human. OpenAI did not identify the editor.

Mathematicians’ Assessment

The mathematical community’s reaction has been positive but cautious. After two weeks of examination, experts largely considered the shorter argument correct. Radziwiłł was able to outline it briefly, other mathematicians circulated longer explanations, and Peter Sarnak made it the subject of a course.

Still, Sarnak said he would not call it a theorem until mathematicians had checked and understood every detail.

How the Proof Came Together

For decades, Sarnak had argued that the zeta function should be studied as part of a family of related functions rather than in isolation. The challenge was identifying the right family.

The proof uses a previously overlooked family involving metaplectic forms, along with existing mathematical results—including a formula proved in 2021 by Alexander Dunn and Radziwiłł.

The breakthrough was largely a synthesis of existing ideas, not a wholly new mathematical concept. It combined established tools in an unexpected way and applied them to a family that experts had not recognized as useful for this problem. The AI also used tools that mathematicians had thought would not be sufficient.

The video’s assessment is that the achievement lay partly in finding the right family and partly in being willing to try existing tools where experts had expected them to fail.

What the Result Does—and Doesn’t—Show

The result does not settle the full Riemann hypothesis. Will Sawin suggested that, even once the new result is fully understood, mathematicians may still be no closer to proving the full hypothesis.

The video’s narrator nevertheless calls it AI’s biggest success in mathematics to date and emphasizes how much it drew on earlier work by human mathematicians.

The result also raises a broader question. Alex Kontorovich said he had struggled to identify something mathematicians can do that AI could never do. The video ends by asking what, if anything, that might be.

Speakers and Sources Featured

• Video narrator — Explains the result and gives the video’s assessment.
• Alex Kontorovich — Mathematician who offered early hypotheses about how the AI reached its result.
• Maksym Radziwiłł — Mathematician who assessed the proof and co-proved a formula used in it.
• Manjul Bhargava — Fields Medalist who relayed colleagues’ early reactions.
• Stephen Wolfram — Suggested the result might come from combining ideas from different places.
• Peter Sarnak — Number theorist whose long-standing view about studying families of functions is relevant to the proof.
• Michael Rubinstein — Commented on the difficulty of choosing the right family.
• Henryk Iwaniec — Studied related families and came close to the one used in the proof.
• Alexander Dunn — Studied metaplectic forms and co-proved the 2021 formula used in the argument.
• Will Sawin — Offered a cautious view about how much the result advances the full Riemann hypothesis.
• Thomas Bloom — Remarked that the result would recently have seemed unattainable.
• Quanta Magazine — Publication reporting on mathematicians’ examination of the proof.
• OpenAI — Publisher of the two proof versions discussed.
• Lean — Proof assistant used to check the longer version.