{"slug": "openai-solves-quasi-riemann-hypothesis-understand-what-it-means", "title": "OpenAI Solves Quasi Riemann Hypothesis: Understand What It Means", "summary": "OpenAI posted a paper titled \"The Quasi-Riemann Hypothesis\" claiming that every Dirichlet L-function, including the Riemann zeta function, has no zeros in the half-plane where the real part of s exceeds 11/12, a result released as part of a dump of 722 mathematical manuscripts from an unreleased internal model. The claim, if correct, squeezes the zeros into a band from roughly 0.083 to 0.917 and extends to Hecke L-functions over K = Q(√-3), but it is a deliberately weaker statement than the Riemann Hypothesis, which remains unproven and carries a $1 million Millennium Prize from the Clay Mathematics Institute. The paper is drawing the most attention from number theorists among the 722 manuscripts.", "body_md": "*A new OpenAI paper claims to have pushed back the boundary on where the zeros of the Riemann zeta function can hide, a problem that has resisted mathematicians for well over a century. It isn’t the Riemann Hypothesis itself, but it may be the biggest step toward it in generations. Here’s what that actually means.*\n\nOpenAI today posted a paper titled “The Quasi-Riemann Hypothesis”. It was released as part of a [giant dump of 722 mathematical manuscripts](https://officechai.com/ai/openai-releases-over-300-mathematical-results-produced-by-an-internal-model/) from an unreleased internal model, and it is already the one drawing the most attention from number theorists. The abstract claims that every Dirichlet L-function, including Riemann’s famous zeta function, has no zeros in the half-plane where the real part of s is greater than 11/12.\n\nIf that sentence means nothing to you, that’s fine. Here’s the plain-English version.\n\n## First, what is the Riemann Hypothesis?\n\nStart with prime numbers: 2, 3, 5, 7, 11, 13 and so on, numbers divisible only by themselves and 1. They look random. They thin out as numbers get bigger, but unpredictably, and nobody has found a simple formula for where the next one lands.\n\nIn 1859, the German mathematician Bernhard Riemann noticed that the primes’ behaviour is tied to a strange function, now called the Riemann zeta function. You can feed it complex numbers (numbers with a “real” part and an “imaginary” part, which you can picture as a point on a flat map). For most inputs it gives some answer. For certain special inputs it gives exactly zero. These inputs are called the function’s “zeros.”\n\nHere’s the connection: where the zeros sit determines how evenly primes are scattered. Each zero contributes a kind of ripple to the count of primes, and the further a zero strays from the ideal position, the bigger and wilder the ripples.\n\nRiemann guessed that all the interesting zeros sit on one perfectly straight vertical line on that map, the line where the real part equals exactly 1/2. That guess is the **Riemann Hypothesis**. Computers have checked trillions of zeros and every one has landed on the line. But no one has proven it, and it carries a $1 million Millennium Prize from the Clay Mathematics Institute.\n\nA rough picture: imagine a vertical strip of map, with its left edge at 0 and its right edge at 1. All the interesting zeros live somewhere inside this strip. The Riemann Hypothesis says they’re all stacked on the dead-centre line. What mathematicians have been able to prove is far weaker: the zeros can’t be on the right edge itself, and can’t be in a thin sliver next to it. But that sliver *shrinks* the higher up you go, so for 160 years nobody could rule out zeros creeping arbitrarily close to the edge.\n\n## So what is the *quasi*-Riemann Hypothesis?\n\nIt’s a deliberately weaker version. Instead of demanding that every zero sit on the middle line, it asks for something less ambitious: that there is some fixed gap, any gap, between the zeros and the right edge of the strip, and that it doesn’t shrink as you climb.\n\nMathematicians have long used this as a stepping stone. It’s the difference between “the zeros are all exactly in the middle” and “the zeros are at least definitely not hugging the edge.”\n\nOpenAI’s paper claims to deliver that, and with real numbers attached. It says there are no zeros anywhere to the right of the line at 11/12, roughly 0.917. Because zeros come in symmetric pairs around the middle line, that squeezes all of them into a band running from about 0.083 to 0.917 instead of the whole strip.\n\nThe paper goes further than the Riemann zeta function. The same result is claimed for every Dirichlet L-function, which are close cousins of zeta used to study primes in arithmetic progressions (for example, how primes are distributed among numbers ending in 1, 3, 7 and 9). It also covers a family of related functions called Hecke L-functions over a particular number system, K = Q(√-3).\n\nOne note on the numbers: the paper shown in OpenAI’s repository states 11/12, while OpenAI’s catalog and much of the coverage cites 7/8 (0.875), a tighter bound. The catalog lists the result as a family of three manuscripts, and a companion paper gives an alternate argument for the 11/12 figure. Either way, the zeros are being pushed well away from the edge.\n\n## The Landau–Siegel zero, and why number theorists care\n\nThe abstract also says the result rules out “Landau–Siegel zeros,” and this may matter to specialists even more than the headline.\n\nA Landau–Siegel zero is a hypothetical, suspicious zero of certain L-functions that would sit extremely close to 1 on the real line. No one has ever found one, and no one has ever been able to prove it doesn’t exist. Its ghostly possibility has haunted analytic number theory for nearly a century. It’s the reason many famous theorems about primes come with constants that mathematicians can prove *exist* but can’t actually calculate.\n\nAn independent summary of the release notes that the exclusion is of the form “no such zeros close enough to 1, with a constant that doesn’t depend on the conductor.” In other words, it removes the ghost where it matters most, though not every conceivable real zero everywhere.\n\n## What does the proof actually do?\n\nAt a high level, according to a summary of the manuscripts, the argument measures the same sum of character values two different ways. One calculation bounds it directly. A second route links it to an integral involving the reciprocal of an L-function. The claimed savings in the estimates let the reciprocal be extended past a hypothetical rightmost zero, which produces a contradiction. If such a zero existed, the two calculations couldn’t agree.\n\nThe paper’s contents list gives a sense of the machinery: Poisson summation, a “dual mean square” estimate that gets iterated, and a long appendix on theta function calculations. This is deep, technical, classical-style analytic number theory, not a brute-force computer search.\n\n## Why it matters\n\n**It could sharpen what we know about primes.** A zero-free region translates directly into how accurately we can count primes. Today’s best proven estimates beat the naive prediction only by a factor that improves very slowly. A fixed zero-free half-plane would give a “power saving,” meaning the error shrinks like a fixed power of the number you’re working with. It still wouldn’t match what the full Riemann Hypothesis would give, but it’s a big upgrade and it would flow into many results about primes in arithmetic progressions.\n\n**It tackles a ghost.** Excluding Landau–Siegel zeros uniformly could turn many “ineffective” theorems, ones that say a bound exists without telling you what it is, into ones with concrete, usable constants.\n\n**It’s not the Riemann Hypothesis.** To be clear, the zeros could still, in principle, wander anywhere inside the band between 0.083 and 0.917. The $1 million prize is not on the table. But experts quoted in coverage of the release appear to treat the result as a major leap, and Levent Alpöge, a mathematician, has praised the quasi-Riemann and no-Siegel-zero results in strong terms.\n\n## The caveat: it isn’t verified yet\n\nEverything here comes with a big asterisk. The paper hasn’t been peer reviewed, and outside mathematicians haven’t confirmed it. OpenAI says it consulted the Institute for Advanced Study’s Advisory Group on Mathematics and AI about how to release the work, and says the model is not public. The paper itself says it was written with human assistance, and OpenAI’s repository notes that the zero-free-region write-up was edited by people for readability.\n\nThe result is reportedly accompanied by a Lean formalization, a machine-checkable version of the proof. That’s a meaningful safeguard, but mathematicians will still want to check that the formal statement says what the informal one claims. And Terence Tao has warned that [AI-generated proofs can look flawless while hiding subtle mistakes](https://officechai.com/ai/ai-can-generate-math-proofs-that-look-flawless-but-makes-subtle-mistakes-that-humans-wouldnt-terrance-tao/). Questions of credit are also getting messier as labs and individuals converge on the same problems, as seen when [OpenAI said it hadn’t seen Buckmaster and Alpöge’s work](https://officechai.com/ai/openai-says-it-didnt-see-any-of-buckmaster-and-alpoges-work-while-resolving-navier-stokes/) while resolving Navier–Stokes.", "url": "https://wpnews.pro/news/openai-solves-quasi-riemann-hypothesis-understand-what-it-means", "canonical_source": "https://officechai.com/ai/quasi-riemann-hypothesis-explanation/", "published_at": "2026-10-07 07:39:32+00:00", "updated_at": "2026-10-07 07:47:43.615911+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "machine-learning"], "entities": ["OpenAI", "Riemann Hypothesis", "Riemann zeta function", "Dirichlet L-function", "Hecke L-function", "Bernhard Riemann", "Clay Mathematics Institute"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/openai-solves-quasi-riemann-hypothesis-understand-what-it-means", "markdown": "https://wpnews.pro/news/openai-solves-quasi-riemann-hypothesis-understand-what-it-means.md", "text": "https://wpnews.pro/news/openai-solves-quasi-riemann-hypothesis-understand-what-it-means.txt", "jsonld": "https://wpnews.pro/news/openai-solves-quasi-riemann-hypothesis-understand-what-it-means.jsonld"}}