内容来源：https://www.quantamagazine.org/ai-proved-the-quasi-riemann-hypothesis-what-happens-now-20261019/

内容总结：

近日，人工智能在数学领域取得一项引发广泛关注的进展。OpenAI于2026年10月6日公布的研究成果中，有一项声称解决了“准黎曼假设”——即黎曼假设的一个较弱版本。黎曼假设被认为是数学界最大的未解难题之一，已悬置160余年。斯坦福大学数学家Jared Duker Lichtman评价称，这是“迄今为止AI在数学领域最重要的成就，且领先幅度相当大”。

纽约大学数学家Maksym Radziwiłł表示，即便只是“准黎曼假设”这一较为有限的结果，其影响也是“惊人的”，“从某种意义上说，黎曼假设的绝大多数推论已经可以从准黎曼假设中得出”。与数学家们长期预期不同的是，该证明并未引入任何新概念，而是揭示了已有概念之间隐藏的联系。佐治亚理工学院数学家Alexander Dunn称：“这个证明是对众所周知的想法的一次令人惊讶的综合。”

该结果部分依赖于Radziwiłł和Dunn此前开创的技术。他们与其他数论学家一直在努力理解这一证明及其意义，目前已开始得出一些答案。

黎曼假设涉及黎曼ζ函数零点的分布。该函数与素数分布密切相关——素数被称为“算术的原子”。黎曼假设断言，ζ函数的所有非平凡零点都位于实部为1/2的垂直直线上。数学家已通过数值计算验证了前12万亿个非平凡零点均位于该线上，但无法保证对所有零点都成立。哪怕出现一个例外，后果都将是地震级的。

“准黎曼假设”只要求证明所有非平凡零点都位于比已知的0到1之间更窄的垂直带内。带越窄，关于素数分布的结论就越精确。OpenAI公布了两个版本的证明：第一版 dated 2026年9月30日，声称证明所有非平凡零点的实部位于1/8到7/8之间，并已用Lean编程语言形式化验证，论文长达199页；第二版约一周后完成，尚未形式化验证，声称证明了较弱的结果——零点实部位于1/12到11/12之间，论文仅49页。Radziwiłł指出，这两篇论文“写得非常糟糕（在任何数学期刊都会立即被拒）”，“在表述和清晰度方面远达不到我们领域的要求”。因此数学家们暂时搁置了7/8的论文，集中研究11/12的版本。

值得注意的是，11/12的论文是OpenAI当天发布的722篇论文中唯一经过人工编辑的一篇。OpenAI未回应关于编辑者身份及为何仅此一篇经人工处理的询问，仅表示“该论文是OpenAI全体人员工作的成果”。

论文发布后的数周内，数学家们对其进行了密集研究。普林斯顿大学知名数论学家Peter Sarnak决定将本学期的一门课程改为专门研究该论文。Radziwiłł表示他现在可以用半页纸勾勒出证明的论证思路，其他数学家撰写的更详细的说明在7到13页之间。专家们大体同意该结果正确，其新颖之处在于将数论中已存在一段时间的思想结合在一起。

数十年来，Sarnak一直主张数学家应将ζ函数视为更广泛函数族中的一个成员，然后证明关于该函数族的命题。滑铁卢大学的Michael Rubinstein表示：“但具体如何做、考虑哪个函数族，并不清楚。”事实证明，Sarnak的直觉是对的，但OpenAI模型识别出的函数族此前被忽视了。罗格斯大学数学家Henryk Iwaniec在2002年曾与之擦肩而过。找到正确的函数族需要一种名为“metaplectic forms”的冷门数学对象，Dunn过去五年一直在研究它。Dunn说：“没人预料到这一点。”他补充说，metaplectic forms此前在数论中并不常见，“所以我希望这个美丽而深奥的理论能得到应有的关注”。

令Dunn和Radziwiłł惊讶的是，他们2021年证明的一个公式成为该证明的关键组成部分。Radziwiłł回顾说，研究这一数论分支的数学家此前错误地认为他们的工具不足以解决准黎曼假设，坚信需要另一分支的技术。AI模型“相信”他们的工具可能足够，这一视角至关重要。Radziwiłł说：“那是一个相当大胆的信念！”

数学家们对OpenAI的工作既感到沮丧又感到兴奋。Sarnak在邮件中表示，在人类检查并理解所有细节之前，他不会满意地称之为“定理”。但Radziwiłł说：“总体感觉是如释重负。”准黎曼假设的证明将一举确立数论、几何和理论计算机科学领域的大量结果。

数论学家对将该结果的技术组合将7/8的界推到诱人的1/2持怀疑态度。普林斯顿大学的Will Sawin表示：“在我看来，最可能的情况是，一旦数学家完全内化了新方法并厘清其含义，我们与完整黎曼假设的距离仍将和之前一样遥远。但我并不确定。”曼彻斯特大学的Thomas Bloom也认为完整黎曼假设仍遥不可及，但他补充说：“几周前我们也会对准黎曼假设说同样的话！到了这个地步，什么都不会让我惊讶了。”

中文翻译：

AI证明了拟黎曼猜想。接下来会发生什么？

引言

在OpenAI于2026年10月6日公布的377项成果中，有一项格外引人注目。它声称解决了“拟黎曼猜想”——这是黎曼猜想的一个较弱版本，而黎曼猜想可以说是数学中最大的未解问题。正如斯坦福大学的贾里德·杜克·利希特曼所说，这一新证明“是迄今为止AI在数学领域最重要的成就，而且领先幅度相当大”。

黎曼猜想作为未解问题已有160多年。解决它将使数学家能够更好地理解素数的本质，而素数是数学中最重要的基本构件之一。而这还仅仅是开始。

纽约大学的马克西姆·拉齐维乌表示，即便是更为有限的“拟黎曼猜想”结果，其影响也是“惊人的”。“从某种意义上说，黎曼猜想绝大多数推论已经可以从拟黎曼猜想中得出。”

此外，与数学家们长期以来的预期相反，这一证明并不需要引入任何新概念——它反而揭示了旧概念之间隐藏的联系。“这个证明是对众所周知的想法的一次令人惊讶的综合，”佐治亚理工学院的亚历山大·邓恩说。

这一结果部分依赖于拉齐维乌和邓恩开创的技术。他们与其他数论学家一起，一直在努力理解这一证明及其可能意味着什么。现在他们开始得到答案了。

从Zeta到更远处

要理解拟黎曼猜想说的是什么，你首先需要理解黎曼猜想。当克雷数学研究所在世纪之交宣布七个千禧年大奖难题时，黎曼猜想赫然居于其余六个之上。它是唯一一个从大卫·希尔伯特一个世纪前宣布的那份著名的纲领性问题清单中延续下来的问题。

从表面上看，这个猜想涉及一个特定函数的行为。从求和 $latex 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots $ 开始。现在用这个和（配合一个特殊的数学技巧来扩展其范围）来构建一个更一般的函数，称为黎曼zeta（ζ）函数：

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

这里，输入s是一个复数。复数有两个部分——一个“实”部和一个“虚”部，虚部是−1的平方根（写作i）的倍数。任何复数都可以被视为平面上的一个点：

马克·贝兰/《量子杂志》

黎曼猜想是关于s取哪些值会使zeta函数等于零。

但它也远不止关乎这个特定函数的行为方式。正如伯恩哈德·黎曼在1859年首次提出这一猜想时所意识到的那样，这个函数与素数中的规律有关——素数是指只能被1和自身整除的整数，许多数学家称之为“算术的原子”。

人们早就知道，沿着数轴移动时，素数会逐渐稀疏。黎曼表明，素数究竟如何稀疏，从某种意义上说，是由zeta函数支配的——特别是由它等于零的位置所决定。

不难证明，如果s是负偶数（−2、−4、−6、……），那么ζ(s) = 0。这些被称为平凡零点。黎曼猜想认为，zeta函数仅有的其他零点——称为非平凡零点——出现在s的实部为1/2时。也就是说，它们应该全部位于一条垂直线上：

如果在远离那条垂直线的某处发现了任何非平凡零点，那就意味着素数必然形成一种极其令人惊讶的规律，一种迄今从未被观察到的规律。

仅仅缩小非平凡零点可能出现的区域就已经产生了巨大的影响。黎曼本人证明了zeta函数的非平凡零点只能出现在0到1之间的一个竖直带中。然后，在1896年，两位数学家证明了零点不可能位于该带的边缘上。作为直接推论，他们得以证明一个重要的结果，称为素数定理——一个描述素数平均稀疏速度的公式。

如果黎曼猜想成立，那么数学家就可以保证，素数定理所给出的平均值之间的任何偏差将始终很小。

数学家已经通过数值计算验证了前12万亿个非平凡零点都在1/2线上。但他们无法确定该猜想对所有零点都成立，哪怕只有一个孤立的例外，也会像地震一样震撼。正如恩里科·邦别里在阐述该问题千禧年大奖版本的论文中所写：“黎曼猜想的失败将在素数分布中造成混乱。”

糟粕中的钻石

这就把我们带到了拟黎曼猜想。黎曼猜想假定所有非平凡零点都在1/2处的垂直线上，而“拟”版本只要求证明它们全部包含在一个比已知的0到1之间的带更窄的竖直带中。带越窄，人们能说的关于素数分布的话就越多。

OpenAI公布了两个不同版本的拟黎曼猜想证明。第一个版本日期为2026年9月30日，声称证明所有非平凡零点必须位于1/8到7/8之间，并使用Lean编程语言进行了形式化验证。第二个版本几乎一周后完成（尚未形式化验证），声称证明了一个较弱的结果：零点的实部必须在1/12到11/12之间。但第一篇文章长达199页，而第二篇相比之下只有区区49页。

与许多近期AI生成的成果一样，这两篇论文“写得非常糟糕（在任何数学期刊都会立即被拒的那种），”拉齐维乌在一封电子邮件中说。“它们远远达不到我们领域在阐述和清晰度方面的标准。”因此数学家们把7/8那篇论文放在一边，专注于11/12那篇，后者似乎涉及类似的技术并有类似的推论。“人们主要关注11/12结果的原因是它更短，”拉齐维乌说。

加州理工学院

加州理工学院

并非巧合的是，11/12的结果是10月6日发布的722篇OpenAI论文中唯一一篇经过人工编辑的。OpenAI尚未回应关于此人是谁、或者为什么这是唯一一篇在发布前得到人类专业知识帮助的论文的询问。在回答《量子杂志》的提问时，该公司表示：“这篇论文是OpenAI所有人共同努力的结果。”

自论文发布以来的几周里，数学家们一直在密集地研究它。普林斯顿大学著名数论学家彼得·萨纳克决定将他本学期教授的一门课程改为研究这篇论文。拉齐维乌说，他现在可以用半页纸概述证明的论证，而其他数学家更详细的阐述则在7到13页之间。（截至本文写作时，这些尚未发表，但正在私下流传。）这些专家大体上同意该结果是正确的——并且其新颖性在于它汇集了数论中已存在一段时间的思想。

几十年来，萨纳克一直主张数学家应该把zeta函数视为更广泛函数族中的一员来研究黎曼猜想，然后证明关于该函数族的命题。“但之前不清楚具体如何做到，或者应该考虑哪个函数族，”滑铁卢大学的迈克尔·鲁宾斯坦说。

事实证明，萨纳克的直觉是对的。但OpenAI模型所识别出的那个函数族此前一直被忽视。罗格斯大学的数学家亨里克·伊万涅茨在2002年与它擦肩而过，当时他为了证明黎曼猜想而识别出了若干密切相关的函数族。

找到正确的函数族需要一种名为“元-plectic形式”的冷僻数学对象，邓恩过去五年一直在研究它。“没有人预料到这一点，”他说。元-plectic形式在此之前在数论中并不常见，他补充说，“所以我希望这个美丽而深奥的理论能得到它应有的关注。”

令邓恩和拉齐维乌惊讶的是，他们在2021年证明的一个公式成为了该证明的关键组成部分（连同众多其他数学家的成果一起）。拉齐维乌回顾说，研究所有这些工作所属的数论分支的数学家们此前错误地认为他们的工具不够强大，无法解决拟黎曼猜想。他们一直确信，证明它需要来自数论另一个完全不同的分支的技术。AI模型“相信”他们的工具可能足够了，这是一个关键的视角。“这是一个相当大胆的信念！”拉齐维乌说。

数学家们对OpenAI的工作既感到沮丧又感到兴奋。萨纳克本人在一封电子邮件中说，在人类检查并理解所有细节之前，他不会满意地称其为“一个定理”。但“总体感觉是如释重负，”拉齐维乌说。不仅有数论领域，还有几何学和理论计算机科学领域的数十项成果，将因拟黎曼猜想的证明而在一举之间被确立为数学真理。

数论学家怀疑，该结果中技术的特定组合能否让他们把7/8的界降到诱人的1/2。“在我看来最可能的是，一旦数学家们完全内化了这些新方法并厘清了它们的推论，我们仍然会感觉离完整的黎曼猜想和之前一样遥远，”普林斯顿的威尔·萨温说。“但我并不确定。”

曼彻斯特大学的数学家托马斯·布鲁姆同意完整的黎曼猜想仍然似乎遥不可及。但他补充说，“几周前我们也会对拟黎曼猜想说同样的话！到了这一步，什么都不会让我惊讶了。”

英文来源：

AI Proved the Quasi-Riemann Hypothesis. What Happens Now?
Introduction
Among the 377 results that OpenAI announced on October 6, 2026, one stood out. It claimed to settle the “quasi-Riemann hypothesis” — a weaker version of the Riemann hypothesis, which is arguably the biggest open question in math. 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.”
The Riemann hypothesis has been an open problem for over 160 years. Resolving it would enable mathematicians to better understand the nature of prime numbers, one of math’s most crucial building blocks. And that’s just for starters.
The implications of even the more limited “quasi-RH” result are “staggering,” said Maksym Radziwiłł of New York University. “In a sense, the vast majority of consequences of RH already follow from quasi-RH.”
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. “The proof is a surprising synthesis of well-known ideas,” said Alexander Dunn of the Georgia Institute of Technology.
The result relied in part on techniques pioneered by Radziwiłł and Dunn. Alongside other number theorists, they have been working to understand the proof and what it could mean. Now they’re starting to get answers.
To Zeta and Beyond
To understand what the quasi-Riemann hypothesis says, you first have to understand the Riemann hypothesis. When the Clay Mathematics Institute announced seven Millennium Prize Problems at the turn of the century, the Riemann hypothesis loomed above the other six. It was the only one carried over from the celebrated agenda-setting list of problems that David Hilbert had announced a century earlier.
On its surface, the conjecture concerns the behavior of a particular function. Start with the sum $latex 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots $. 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:
$latex \zeta (s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots$
Here, the input s is a complex number. 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). Any complex number can be thought of as a point on a plane:
Mark Belan/Quanta Magazine
The Riemann hypothesis is about which values of s will make the zeta function equal zero.
But it is also about far more than how this particular function behaves. 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.”
It’s long been known that as you move along the number line, the primes thin out. 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.
It’s straightforward to prove that if s is a negative even number (−2, −4, −6, …), then ζ(s) = 0. These are called trivial zeros. 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. That is, they should all be located on a vertical line:
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.
Just narrowing the area where the nontrivial zeros can appear has had huge ramifications. Riemann himself proved that nontrivial zeros of the zeta function can only be found in a vertical strip between zero and 1. Then, in 1896, two mathematicians showed that zeros could not lie on the edges of the strip. 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.
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.
Mathematicians have numerically verified that the first 12 trillion nontrivial zeros are on the 1/2 line. But they can’t be sure the hypothesis holds for all the zeros, and even one solitary outlier would land like an earthquake. 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.”
Diamonds in the Slop
This brings us to the quasi-Riemann hypothesis. 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. The narrower the strip, the more one can say about how the primes are distributed.
OpenAI announced two different versions of their quasi-Riemann hypothesis proof. 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. 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. But while the first paper is 199 pages long, the second weighs in at a comparatively modest 49 pages.
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. “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. “The reason why people look mostly at the 11/12 result is that it’s shorter,” Radziwiłł said.
Caltech
Caltech
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. 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. In response to questions from Quanta, the company said, “The paper is the result of work from everyone at OpenAI.”
Mathematicians have been studying the paper intensely in the couple of weeks since its release. Peter Sarnak, a prominent number theorist at Princeton University, has decided to repurpose a course he’s teaching this semester to study it. 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. (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.
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. “But it was not clear exactly how to do so, or which family to consider,” said Michael Rubinstein of the University of Waterloo.
As it turns out, Sarnak had the right intuition. But the family that OpenAI’s model identified had previously been overlooked. 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.
Finding the right family required obscure mathematical objects called metaplectic forms, which Dunn has spent the past five years studying. “Nobody expected this,” he said. 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.”
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). 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. They’d been convinced that proving it would require techniques from another branch of number theory entirely. The AI model’s “belief” that their tools might be enough was a crucial perspective. “That was a pretty bold belief to have!” Radziwiłł said.
Mathematicians are both frustrated and excited by OpenAI’s work. 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. But “the general feeling is one of relief,” Radziwiłł said. 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.
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. “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. “But I am not certain of this.”
Thomas Bloom, a mathematician at the University of Manchester, agreed that the full Riemann hypothesis still seems out of reach. But, he added, “we would have said the same about quasi-RH a few weeks ago! At this point nothing would surprise me.”