AI Proved the Quasi-Riemann Hypothesis. What Happens Now?

The Riemann hypothesis is a fundamental statement about the distribution of prime numbers.

黎曼假设是关于素数分布的基本陈述。

Among the 377 results that OpenAI announced on October 6, 2026, one stood out.

在 OpenAI 于 2026 年 10 月 6 日公布的 377 项结果中，有一项脱颖而出。

It claimed to settle the “quasi-Riemann hypothesis” — a weaker version of the Riemann hypothesis, which is arguably the biggest open question in math.

它声称解决了“准黎曼假设”——黎曼假设的较弱版本，这可以说是数学中最大的悬而未决的问题。

The new proof is, as Jared Duker Lichtman of Stanford University put it, “the most significant AI-for-math achievement so far, by quite a margin.”

正如斯坦福大学的 Jared Duker Lichtman 所说，新的证明是“迄今为止最重要的人工智能数学成就，遥遥领先”。

The Riemann hypothesis has been an open problem for over 160 years.

160 多年来，黎曼猜想一直是一个悬而未决的问题。

Resolving it would enable mathematicians to better understand the nature of prime numbers, one of math’s most crucial building blocks.

解决这个问题将使数学家能够更好地理解素数的本质，素数是数学最重要的组成部分之一。

And that’s just for starters.

这只是初学者。

The implications of even the more limited “quasi-RH” result are “staggering,” said Maksym Radziwiłł of New York University.

纽约大学的马克西姆·拉齐维乌 (Maksym Radziwił) 表示，即使是更有限的“准 RH”结果，其影响也是“令人震惊的”。

“In a sense, the vast majority of consequences of RH already follow from quasi-RH.”

“从某种意义上说，RH 的绝大多数后果已经是准 RH 的结果。”

Moreover, contrary to what mathematicians had long expected, the proof didn’t need to introduce any new concepts — it instead unearthed hidden connections between old ones.

此外，与数学家长期以来的预期相反，证明不需要引入任何新概念，而是挖掘旧概念之间隐藏的联系。

“The proof is a surprising synthesis of well-known ideas,” said Alexander Dunn of the Georgia Institute of Technology.

“证据是众所周知的想法的令人惊讶的综合，”佐治亚理工学院的亚历山大·邓恩说。

The result relied in part on techniques pioneered by Radziwiłł and Dunn.

这一结果在一定程度上依赖于拉齐维乌和邓恩开创的技术。

Alongside other number theorists, they have been working to understand the proof and what it could mean.

与其他数论学家一起，他们一直在努力理解这个证明及其可能意味着什么。

Now they’re starting to get answers.

现在他们开始得到答案。

To understand what the quasi-Riemann hypothesis says, you first have to understand the Riemann hypothesis.

要理解拟黎曼假设的内容，首先必须了解黎曼假设。

When the Clay Mathematics Institute announced seven Millennium Prize Problems at the turn of the century, the Riemann hypothesis loomed above the other six.

当克莱数学研究所在世纪之交宣布七个千禧年奖问题时，黎曼猜想赫然耸立在其他六个问题之上。

It was the only one carried over from the celebrated agenda-setting list of problems that David Hilbert had announced a century earlier.

这是戴维·希尔伯特一个世纪前宣布的著名议程设定问题清单中唯一遗留下来的问题。

On its surface, the conjecture concerns the behavior of a particular function.

从表面上看，该猜想涉及特定函数的行为。

Start with the sum $latex 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots $.

从 $latex 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots $ 之和开始。

Now use that sum (together with a special mathematical trick to extend its scope) to build a more general function, called the Riemann zeta (ζ) function:

现在使用该和（与扩展其范围的特殊数学技巧一起）构建一个更通用的函数，称为黎曼 zeta (z) 函数：

$latex \zeta (s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots$

$乳胶 \zeta (s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots$

Here, the input s is a complex number.

这里，输入 s 是一个复数。

Complex numbers have two parts — a “real” part and an “imaginary” part, which is a multiple of the square root of −1 (written as i ).

复数有两部分——“实”部分和“虚”部分，“虚”部分是 −1 平方根的倍数（写为 i ）。

Any complex number can be thought of as a point on a plane:

任何复数都可以被视为平面上的一个点：

The Riemann hypothesis is about which values of s will make the zeta function equal zero.

黎曼假设是关于哪些 s 值会使 zeta 函数为零。

But it is also about far more than how this particular function behaves.

但它的意义远不止于此特定函数的行为方式。

As Bernhard Riemann realized when he first made the conjecture in 1859, the function relates to patterns in the prime numbers — integers divisible only by 1 and themselves that many mathematicians call the “atoms of arithmetic.”

正如伯恩哈德·黎曼 (Bernhard Riemann) 在 1859 年首次提出猜想时所意识到的那样，该函数与素数中的模式有关——只能被 1 整除的整数及其本身，许多数学家称之为“算术原子”。

It’s long been known that as you move along the number line, the primes thin out.

人们早就知道，当你沿着数轴移动时，素数会变薄。

Riemann showed that exactly how the primes thin out is, in a sense, governed by the zeta function — in particular by the places where it equals zero.

黎曼证明，从某种意义上说，素数的稀疏程度是由 zeta 函数决定的，特别是在它等于 0 的地方。

It’s straightforward to prove that if s is a negative even number (−2, −4, −6, …), then ζ( s ) = 0.

很容易证明，如果 s 是负偶数 (−2, −4, −6, …)，则 ze( s ) = 0。

These are called trivial zeros.

这些称为平凡零点。

The Riemann hypothesis says that the only other zeros of the zeta function, called nontrivial zeros, occur when s has a real part of 1/2.

黎曼假设指出，zeta 函数的唯一其他零点（称为非平凡零点）出现在 s 的实部为 1/2 时。

That is, they should all be located on a vertical line:

也就是说，它们应该都位于一条垂直线上：

If any nontrivial zero were to be found somewhere away from that vertical line, it would imply that the primes must form an extremely surprising pattern, one that has never been observed to date.

如果在远离垂直线的地方发现任何非平凡的零，则意味着素数必须形成一种极其令人惊讶的模式，一种迄今为止从未被观察到的模式。

Just narrowing the area where the nontrivial zeros can appear has had huge ramifications.

仅仅缩小非平凡零点出现的区域就会产生巨大的影响。

Riemann himself proved that nontrivial zeros of the zeta function can only be found in a vertical strip between zero and 1.

黎曼本人证明了 zeta 函数的非平凡零点只能在 0 和 1 之间的垂直条带中找到。

Then, in 1896, two mathematicians showed that zeros could not lie on the edges of the strip.

然后，在 1896 年，两位数学家证明零不能位于条带的边缘。

As a direct consequence, they were able to prove an important result called the prime number theorem, a formula for how quickly, on average, the primes thin out.

直接的结果是，他们能够证明一个重要的结果，称为素数定理，这是一个计算素数平均变稀的速度的公式。

The connection between the zeta function and the primes was known before Riemann.

zeta 函数和素数之间的联系在黎曼之前就已为人所知。

In 1737, Leonhard Euler showed that for any complex number s , the zeta function is equivalent to another function that can be written entirely in terms of primes:

1737 年，莱昂哈德·欧拉 (Leonhard Euler) 证明，对于任何复数 s ，zeta 函数等价于另一个可以完全用素数编写的函数：

$latex \zeta(s) = \frac{1}{1-\frac{1}{2^s}} \times \frac{1}{1-\frac{1}{3^s}} \times \frac{1}{1-\frac{1}{5^s}} \times \cdots$

$乳胶 \zeta(s) = \frac{1}{1-\frac{1}{2^s}} \times \frac{1}{1-\frac{1}{3^s}} \times \frac{1}{1-\frac{1}{5^s}} \times \cdots$

It’s the kind of weird connection that makes mathematicians feel as if they’re accessing a hidden reality.

这种奇怪的联系让数学家感觉好像他们正在接近一个隐藏的现实。

Mathematicians have been building on that crucial link ever since.

从那时起，数学家们就一直在这一关键联系的基础上进行研究。

If the Riemann hypothesis holds, then mathematicians can guarantee that any deviations from the average given by the prime number theorem will always be small.

如果黎曼假设成立，那么数学家可以保证与素数定理给出的平均值的任何偏差总是很小。

Mathematicians have numerically verified that the first 12 trillion nontrivial zeros are on the 1/2 line.

数学家已经通过数值验证，前 12 万亿个非平凡零位于 1/2 线上。

But they can’t be sure the hypothesis holds for all the zeros, and even one solitary outlier would land like an earthquake.

但他们不能确定这一假设适用于所有零，甚至一个孤立的异常值也会像地震一样发生。

As Enrico Bombieri wrote in the paper laying out the Millennium Prize formulation of the problem: “The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers.”

正如恩里科·邦别里（Enrico Bombieri）在阐述该问题的千年奖表述的论文中所写的那样：“黎曼假设的失败将对素数的分布造成严重破坏。”

This brings us to the quasi-Riemann hypothesis.

这给我们带来了拟黎曼假设。

Where the Riemann hypothesis posits that all the nontrivial zeros are on the vertical line at 1/2, the “quasi” version just asks one to prove that they are all contained in a vertical strip that’s narrower than the known strip between zero and 1.

黎曼假设假定所有非平凡零点都位于 1/2 处的垂直线上，而“准”版本只是要求证明它们全部包含在比已知的 0 和 1 之间的条带更窄的垂直条带中。

The narrower the strip, the more one can say about how the primes are distributed.

条带越窄，人们就越能了解素数的分布情况。

OpenAI announced two different versions of their quasi-Riemann hypothesis proof.

OpenAI 宣布了其拟黎曼假设证明的两个不同版本。

The first version, dated September 30, 2026 , claimed to show that all nontrivial zeros must be between 1/8 and 7/8, and was formally verified using the Lean programming language.

第一个版本于 2026 年 9 月 30 日发布，声称所有非平凡零必须在 1/8 到 7/8 之间，并使用精益编程语言进行了正式验证。

The second version, which was completed almost a week later (and is not yet formally verified), claimed to prove a weaker result: that the zeros must have a real part between 1/12 and 11/12.

第二个版本在大约一周后完成（尚未正式验证），声称证明了一个较弱的结果：零必须在 1/12 和 11/12 之间有实部。

But while the first paper is 199 pages long, the second weighs in at a comparatively modest 49 pages.

不过，虽然第一篇论文有 199 页长，但第二篇论文的篇幅相对较小，只有 49 页。

As with many recent AI-generated results, both papers “are horribly written (in a way that would lead to immediate rejection at any mathematical journal),” Radziwiłł said in an email.

与最近许多人工智能生成的结果一样，这两篇论文“写得非常糟糕（这会导致任何数学期刊立即拒绝），”Radziwiłł 在一封电子邮件中说。

“They fall short of the standard of exposition and clarity in our area by a wide margin.” And so mathematicians have set aside the 7/8 paper to focus on the 11/12 one, which seems to involve similar techniques and has similar consequences.

“它们在阐述和清晰度方面远远达不到我们领域的标准。”因此，数学家们把 7/8 的论文放在一边，专注于 11/12 的论文，这似乎涉及类似的技术并产生类似的结果。

“The reason why people look mostly at the 11/12 result is that it’s shorter,” Radziwiłł said.

“人们主要关注 11/12 结果的原因是它更短，”Radziwiłł 说。

Maksym Radziwiłł (top) and Alexander Dunn were surprised that OpenAI’s recent proof of the quasi-Riemann hypothesis used techniques they’d developed years earlier.

Maksym Radziwiłł（上）和 Alexander Dunn 对 OpenAI 最近对准黎曼假设的证明使用了他们多年前开发的技术感到惊讶。

Maksym Radziwiłł (left) and Alexander Dunn were surprised that OpenAI’s recent proof of the quasi-Riemann hypothesis used techniques they’d developed years earlier.

Maksym Radziwiłł（左）和 Alexander Dunn 对 OpenAI 最近对准黎曼假设的证明使用了他们多年前开发的技术感到惊讶。

Not coincidentally, the 11/12 result is the only one of the 722 OpenAI papers released on October 6 to have been edited by a person.

并非巧合的是，11/12 的结果是 10 月 6 日发布的 722 篇 OpenAI 论文中唯一一篇由个人编辑的论文。

OpenAI has not addressed inquiries about who that person is, or why this is the only paper to have had the benefit of human expertise before being released.

OpenAI 没有回答有关此人是谁的问题，也没有回答为什么这是唯一一篇在发布之前受益于人类专业知识的论文。

In response to questions from Quanta , the company said, “The paper is the result of work from everyone at OpenAI.”

在回答广达的提问时，该公司表示：“这篇论文是 OpenAI 全体员工共同努力的成果。”

Mathematicians have been studying the paper intensely in the couple of weeks since its release.

自这篇论文发表以来的几周内，数学家们一直在对其进行深入研究。

Peter Sarnak , a prominent number theorist at Princeton University, has decided to repurpose a course he’s teaching this semester to study it.

普林斯顿大学著名数论学家彼得·萨纳克 (Peter Sarnak) 决定重新调整他本学期教授的课程来研究它。

Radziwiłł says he can now sketch the proof’s arguments in half a page, while more involved write-ups by other mathematicians range from seven to 13 pages.

Radziwiłł 表示，他现在可以在半页内勾勒出证明的论点，而其他数学家的更多涉及的文章则有 7 到 13 页。

(As of this writing, these have not yet been published, but are circulating privately.) These experts largely agree that the result is correct — and that its novelty lies in the way it brings together ideas from number theory that have been around for some time.

（在撰写本文时，这些内容尚未发表，但正在私下流传。）这些专家基本上都认为结果是正确的，其新颖之处在于它汇集了已经存在一段时间的数论思想。

For decades, Sarnak has argued that mathematicians should approach the Riemann hypothesis by considering the zeta function as just one member of a broader family of functions, and then proving statements about that family.

几十年来，萨纳克一直认为，数学家应该通过将 zeta 函数视为更广泛的函数族中的一个成员来研究黎曼假设，然后证明有关该族的陈述。

“But it was not clear exactly how to do so, or which family to consider,” said Michael Rubinstein of the University of Waterloo.

滑铁卢大学的迈克尔·鲁宾斯坦 (Michael Rubinstein) 表示：“但目前尚不清楚具体如何做到这一点，也不清楚应该考虑哪个家庭。”

As it turns out, Sarnak had the right intuition.

事实证明，萨纳克的直觉是正确的。

But the family that OpenAI’s model identified had previously been overlooked.

但 OpenAI 模型识别的家族此前曾被忽视。

The Rutgers University mathematician Henryk Iwaniec narrowly missed it in 2002, when he identified a number of closely related families in an effort to prove the Riemann hypothesis.

2002 年，罗格斯大学数学家亨利克·伊万尼克 (Henryk Iwaniec) 差点错过了这一机会，当时他确定了一些密切相关的家族，试图证明黎曼猜想。

Video : Alex Kontorovich, professor of mathematics at Rutgers University, breaks down the notoriously difficult Riemann hypothesis in this comprehensive explainer.

视频：罗格斯大学数学教授 Alex Kontorovich 在这个全面的解释中打破了众所周知的困难黎曼假设。

Finding the right family required obscure mathematical objects called metaplectic forms, which Dunn has spent the past five years studying.

找到合适的家族需要被称为元波形式的晦涩数学对象，邓恩在过去的五年里一直在研究它。

“Nobody expected this,” he said.

“没有人预料到会这样，”他说。

Metaplectic forms had been somewhat uncommon in number theory until now, he added, “so I hope this beautiful and esoteric theory gets the attention it deserves.”

他补充道，到目前为止，元合形式在数论中还不太常见，“所以我希望这个美丽而深奥的理论得到应有的关注。”

To Dunn and Radziwiłł’s surprise, a formula they had proved in 2021 became a crucial ingredient in the proof (along with the work of numerous other mathematicians).

令 Dunn 和 Radziwiłł 惊讶的是，他们在 2021 年证明的一个公式（以及许多其他数学家的工作）成为证明中的关键要素。

In retrospect, Radziwiłł said, mathematicians who study the branch of number theory that all this work belongs to were mistaken in thinking that their tools weren’t strong enough to resolve the quasi-Riemann hypothesis.

拉齐维乌说，回想起来，研究所有这些工作所属的数论分支的数学家错误地认为他们的工具不够强大，无法解决拟黎曼假设。

They’d been convinced that proving it would require techniques from another branch of number theory entirely.

他们坚信，证明这一点需要完全来自数论另一个分支的技术。

The AI model’s “belief” that their tools might be enough was a crucial perspective.

人工智能模型“相信”他们的工具可能足够了，这是一个至关重要的观点。

“That was a pretty bold belief to have!” Radziwiłł said.

“这是一个非常大胆的信念！”拉齐维乌说道。

Mathematicians are both frustrated and excited by OpenAI’s work.

数学家们对 OpenAI 的工作既感到沮丧又兴奋。

Sarnak, for his part, said in an email that he won’t be satisfied that it’s “a Theorem” until humans check and understand all the details.

萨纳克则在一封电子邮件中表示，在人类检查并理解所有细节之前，他不会满意这是一个“定理”。

But “the general feeling is one of relief,” Radziwiłł said.

但“总体感觉是松了一口气，”拉齐维乌说。

There are scores of results across not just number theory but also geometry and theoretical computer science that the proofs of quasi-Riemann will, in a swift stroke, establish as mathematical truths.

不仅在数论中，而且在几何学和理论计算机科学中都有大量的结果，拟黎曼的证明将很快将它们确立为数学真理。

Number theorists are skeptical that the result’s particular combination of techniques will let them take the 7/8 bound down to the tantalizing 1/2.

数论学家怀疑该结果的特定技术组合能否让他们将 7/8 降到诱人的 1/2。

“It seems most likely to me that once mathematicians have fully internalized the new methods and worked through their implications, we will still feel as far off from the full Riemann hypothesis as we were before,” said Will Sawin of Princeton.

普林斯顿大学的威尔·萨温说：“在我看来，一旦数学家完全内化了新方法并研究了它们的含义，我们仍然会像以前一样感觉距离完整的黎曼假设还很遥远。”

“But I am not certain of this.”

“但我对此并不确定。”

Thomas Bloom , a mathematician at the University of Manchester, agreed that the full Riemann hypothesis still seems out of reach.

曼彻斯特大学数学家托马斯·布鲁姆（Thomas Bloom）也认为，完整的黎曼假设似乎仍然遥不可及。

But, he added, “we would have said the same about quasi-RH a few weeks ago!

但是，他补充道，“几周前我们也会对准 RH 说同样的话！

At this point nothing would surprise me.”

在这一点上，没有什么会让我感到惊讶的。”

Get highlights of the most important news delivered to your email inbox

获取发送到您的电子邮件收件箱的最重要新闻的摘要

Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation.

《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。

Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.

辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。

Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.

版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。