张益唐苦攻数十年的Siegel零点难题，竟被攻破了！一道困扰数论界近百年的难题，如今被浓缩成了一份仅有5页的证明。2007年、2022年，张益唐两次发论文冲击它，最后那篇写了111页，拿下的也只是一个弱化版本。就在这两天，牛津数学家Ben Green甩出了一份排除Siegel零点的证明，在数学圈炸开了锅。而这5页，竟然出自Green和ChatGPT 6 Astra的一次「vibemathing」，也就是和AI边聊边把证明做出来，他事后只是亲手把论证重写了一遍。在论文首页，他写下了一句：「本文不主张任何人类原创性。」论文地址：https://hexagonmath.org/pdf/2610.00311v1当年和陶哲轩联手证明「素数里藏着任意长的等差数列」的，正是他。就在10月6日，OpenAI一个连名字都没公布的内部模型，率先攻下了这道难题。
此后，数学家们带着AI接力，只用了12天，就把证明削成了5页。近百年里，从Siegel到张益唐，一代代数学家在这道题前止步。这一次，AI率先撞开了大门，人类数学家紧跟着冲了进去。10月6日，OpenAI一口气公开了700多篇数学手稿。在这堆大部头中间，夹着一篇很薄的论文《一致排除Landau–Siegel零点》，正文只有8页。论文地址：https://github.com/openai/math仅仅5天后，密苏里大学数学家William Banks就接过了第一棒。他把论证从头铺开，写成19页的详尽版，还把原证明里只知道「存在」的那个常数，算成了一个具体的数：0.003。论文地址：https://arxiv.org/abs/2610.15483又过了4天，澳大利亚新南威尔士大学堪培拉分校的Bryce Kerr，一刀把证明砍到了6页。
他把原证明里最繁琐的插值和行列式推导，整个换成了「鸽巢原理」，也就是10个苹果放进9个抽屉，总有一个抽屉里不止一个苹果。Kerr在致谢里写明，ChatGPT参与了推导和写作。论文地址：https://arxiv.org/abs/2610.2109910月18日，斯坦福大学数学家Jared Lichtman激动地转发了Kerr的论文，直呼它「也许就是那本书里的证明」。传奇数学家Erdős常说，上帝有一本书，记着每个定理最漂亮的证明。Lichtman随即发帖，把这一幕比作1954年班尼斯特第一个跑进4分钟一英里。数学家研究素数分布，靠的是一类叫L函数的工具。它的零点离1越远，素数就分得越均匀，偏偏有些L函数可能藏着一个紧贴着1的实数零点。这就是Siegel零点。它像一个幽灵，谁也没见过，谁也证明不了它不存在。近百年来，大量关于素数分布的定理，都只能拖着一条尾巴：「除非存在Siegel零点」。2022年时，张益唐那篇论文给出的界，离彻底赶走幽灵还差着一大截。
而OpenAI这次，则证明了这种紧贴着1的零点根本不存在。也就是说，所有零点和1之间，都至少隔着一段距离，幽灵从此无处可藏。
这虽然还不是黎曼猜想，但对研究经典解析数论的人来说，这个幽灵卡了好几代人。Green的5页，走的是反证法。假设幽灵存在，就能逼出一个不可能存在的整数。如果幽灵存在，绝大多数小素数都会被迫站到同一边。Green用这些素数造出一个巨大的正整数det G，它的大小有一个天花板。关键一步像照镜子。对这些素数取p次方，√d就变成了−√d，det G也因此被它们一层层整除，至少得越来越大。一边是天花板，一边是越抬越高的地板。地板撞穿天花板，矛盾出现，幽灵消失。比起OpenAI原版，Green全程只待在一个更小的数域里，行列式也直接写成了显式的式子。这场冲击，直接砸到了Green自己头上。6月，他刚拿下欧洲研究理事会的高级资助。4个月后，大约四分之三的研究目标就被AI攻破了。他在数学社区Proofs and Prompts的集体征文里写道：
我没料到其中一些成果会有这么大的量级，这让我震惊极了。
想法一点也不缺。某种程度上，正因为量级太大，压力反而没了。
陶哲轩就没这么洒脱了。他痛心的是，这些AI证明背后没有一个人能回答问题、做报告、带学生，没人能参与后续发展。不到两周，后续发展真的来了，只是AI依然在场。Kerr的致谢里写着ChatGPT，Green干脆在第一页声明，不主张任何人类原创性。接过这份证明的人类数学家，干的活变成了把AI的证明读懂、改写，再讲给更多人听。一位证明过格林-陶定理的顶级数学家，成了AI证明的重写者。放在一年前，没人敢想这个画面。10月20日，Banks还要去德国波恩的马普数学所做两场报告，把这份证明从头到尾讲给同行听。这道从1935年悬到现在的难题，就这样从AI的手稿，走进了数学家的课堂。伦敦玛丽女王大学数学教授Abhishek Saha看完这几版证明说，研究经典解析数论的人在这道题上卡了很多年，大家都觉得它难得离谱。他忍不住想，还有多少解析数论的猜想，也能被这样短短几页纸拿下。放在以前，这种事只能等哪个天才灵光一闪。现在，AI先把路蹚出来，数学家再拉着AI把路修短、修直，后面的人就都能跟上。以前最难的是找到证明，现在难的是读懂它。就连读懂这一步，AI也在帮人类提速。下一道百年难题倒下的时候，数学家们已经知道该怎么接棒了。
参考资料：
https://x.com/ObhishekSaha/status/2111862745813455229
https://x.com/thomasfbloom/status/2111718523378499586
编辑：摩西 桃子
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Zhang Litang's decades of Siegel's zero-point problem was broken! A difficult problem that has plagued the digital community for almost a century is now condensed into a five-page certificate. In 2007 and 2022, Zhang Litang published two papers that hit it, concluding with 111 pages and a weakened version. Just these two days, Oxford mathematician Ben Green threw out a zero-point certificate to exclude Siegel and blew up the pot in the math circles. And the five pages came from Green and ChatGPT 6 Astra's "vibemathing" -- that is, talking to AI -- and he then rewritten the argument with his own hands. On the front page of the paper, he wrote: "This paper does not advocate any human originality. Thesis address: https://hexagonmath.org/pdf/261,0.00311v1, in association with Todgeo-chul, confirms that he was the first to attack the problem on October 6, when Open AI had an internal model that had not even been named.
After that, mathematicians with the AI relay took only 12 days to cut the certificate to five pages. For almost 100 years, from Siegel to Zhang Litang, generations of mathematicians have stopped working on the subject. This time, AI broke the door first and human mathematicians came in. On 6 October, Open AI published over 700 mathematical manuscripts in one breath. In the middle of this pile, there is a thin paper, " Unanimously excluding Landau-Siegel Zero ", with only eight pages in the body. Thesis address: Only five days after https://github.com/openai/math, University of Missouri mathematician William Banks took the first hit. He took the argument out of his head and wrote it in a full 19-page version, adding the constant in the original certificate, which only knew " existed ", to a specific number: 0.003. Thesis address: https://arxiv.org/abs/2610.15483, four days later, Bryce Kerr, University of New South Wales, Canberra, Australia, cut the certificate to six pages.
He replaced the original certificate with the most cumbersome plug-in and a horde, which was the Dove Nest principle, with 10 apples in nine drawers, and one drawer with more than one apple. Kerr wrote in his thanks that ChatGPT had been involved in leading and writing. Thesis address: https://arxiv.org/abs/2610.21099, 18 October, University of Stanford Mathematics, Jared Lichtman, passionately transmitted Kerr's paper, calling it “perhaps proof in the book”. Legendary mathematicians, Erdős, used to say that God has a book with the most beautiful proof of every theorem. Lichtman then posted a post comparing the scene to Bannister ' s first four minutes and a mile in 1954. Mathematicians study the distribution of prime numbers by a type of tool called L function. Its zero points are further away from one, and the more even the prime numbers are divided, the more certain L functions may hide a real zero point close to one. That's Siegel zero. It's like a ghost that no one has ever seen and no one can prove it doesn't exist. Over the past 100 years, a great deal of the theorem about the distribution of prime numbers has dragged only one tail: "unless Siegel does not exist." In 2022, Zhang Litang's paper gave a great deal of meaning to banish ghosts.
And Open AI this time proves that this one-point zero doesn't exist. In other words, there is at least a distance between point zero and point one, and there is no place for ghosts to hide.
This is not Lehman's conjecture, but it's a ghost that's been stuck for generations to those who study classic solvency. Green, five pages, go to counter-proof. Assuming a ghost exists, an integer cannot exist. If ghosts exist, the vast majority of small numbers will be forced to stand on the same side. Green uses these numbers to create a huge positive integer det G, the size of which has a ceiling. The key step is like looking in the mirror. Taking the p side of these prime numbers turns √d to −d, and as a result, det G has been divided by at least one layer of them, which is growing. On the one hand, the ceiling, on the other, the floor is raised. The floor went through the ceiling, there was a contradiction and the ghost disappeared. Green stays in a smaller digital area than the original Open AI version, and it is written directly into a visible pattern. The shock hit Green himself. In June, he had just received high-level funding from the European Research Council. Four months later, about three quarters of the research targets were breached by AI. In his collective essay on the mathematics community Proofs and Prompts:
I didn't expect some of these results to be so big, and that shocked me.
There's nothing missing. In a way, the pressure is gone because of the size.
Doo Cheol-chul didn't do that. He lamented the fact that none of those AI certificates could answer questions, report, bring students, and no one could be involved in subsequent development. In less than two weeks, the follow-up has come, and AI is still present. Kerr's thanks contain ChatGPT, and Green simply stated on the first page that he did not advocate any human originality. The work of human mathematicians who took the certificate turned into reading, rewriting and telling more people about the AI certificate. A top mathematician who proved the Green-Tao Theorem, became the rewriter of the AI certificate. A year ago, no one dared to think about it. On 20 October, Banks also went to two reports on Maple mathematics in Bonn, Germany, to present the certificate to his peers from beginning to end. It's a problem from 1935 to the present, so I walked into the mathematicians' class from the AI manuscript. Abhishek Saha, Professor of Mathematics at Queen Mary's University in London, has read these editions to prove that those who study the classic solvency theory have been stuck on the subject for many years, and that it is thought to be very difficult. He couldn't help but wonder how many of the deciphering theories he had been able to take down on such short pages. In the old days, this thing could only wait for some genius to shine. Now, AI gets out of the road, mathematicians pull it down, fix it up, and the people behind you can follow. The hardest thing before was to find proof, and now it is to read it. Even reading this step, AI is helping people speed up. Mathematicians already know how to do it when the next 100-year dilemma falls.
References:
https://x.com/ObhishekSaha/status/2111862745813455229
https://x.com/thomasfbloom/status/2111718523378499586
Editor: Moses Peach
Auto-fetched and translated by AI Atlas. All rights belong to the original author.