[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"news-slug-claude-riemann-zeta-67-percent-record":3,"news-related-db4ffdac-3734-41a2-8e32-67feaa7341bd":38},{"id":4,"title":5,"summary":6,"content":7,"original_url":8,"source_id":9,"tags":10,"translations":24,"news_slug":31,"published_at":32,"created_at":33,"modified_at":34,"is_published":35,"publish_type":36,"image_url":14,"view_count":37},"db4ffdac-3734-41a2-8e32-67feaa7341bd","Claude 冲击黎曼猜想\"失败\",却顺手改写了 37 年没人动过的数学纪录","Anthropic 用未发布的研究版 Claude 在 Claude Code 里冲击黎曼猜想:整体尝试没有解开猜想,但把 zeta 函数零点在临界线上的已证明比例从 41.6% 推进到 67.2%,刷新人类数学家 37 年仅推进 0.8 个百分点的纪录。证明经 Lean 形式化机器验证并由外部数论学家审阅,全程消耗 3100 万输出 token、650 个候选思路全部失败后才命中。","## 一次\"失败\"的 54 小时,顺手改写了数论教科书的一行\n\n先把结论说清楚:Claude 没有证明黎曼猜想。这个自 1859 年悬置至今、挂着克雷研究所百万美元悬赏的难题依然开放。但 8 月 10 日,Anthropic 公布了一个让数论学界集体侧目的结果——一个未发布的研究版 Claude 在 Claude Code 智能体框架里跑了大约一天半,把黎曼 zeta 函数零点落在临界线上的**已被证明比例,从 41.6% 推进到了 67.2%**([Forbes 报道](https:\u002F\u002Fwww.forbes.com\u002Fsites\u002Fjonmarkman\u002F2026\u002F08\u002F13\u002Fclaude-just-broke-a-math-record-that-stood-for-37-years\u002F))。\n\n这个数字是什么概念?此前 37 年里,人类数学家在这个纪录上的全部进展是 0.8 个百分点。Claude 这一跑,单个结果推进了 25.6 个百分点——这是该问题 165 年历史上最大的一次单点跃进。Menlo Ventures 合伙人 Deedy Das 的评价是:这是自 2013 年张益唐有界素数间隔以来,解析数论领域最重磅的结果。\n\n## 它是怎么做到的:31M token、60 个子代理、650 次死胡同\n\n这次运行的工程细节本身就是一份值得拆解的样本(以下数据来自 Forbes 对 Anthropic 论文的转述):\n\n- **两次会话共输出 3100 万 token**,部署了 60 个子代理,执行了 2400 条 shell 命令;\n- 第一个会话生成了 **650 个候选思路,全部是死路**;\n- 出证明的那个会话跑了约一天半,期间不断用数千个已知 zeta 零点做数值校验。\n\n第一场 650 连败,第二场破纪录——这就是把\"失败\"按 token 计价之后的研究范式。\n\n方法论上,Claude 做的是一次教科书级的**跨流派综合**:一条线是 Baluyot、Goldston、Suriajaya、Turnage-Butterbaugh 等人改造 Montgomery 经典技巧的系列论文(使其不再依赖黎曼猜想成立),另一条是 Bombieri 2000 年的工作。两条线都在文献里躺了很多年,只是从没有人把它们\"放在一起读\"。数论分工细到一个人的职业生涯可以完全装进一个子领域,而 Claude 的优势恰恰是**没有专业壁垒——它一次性读完了全部文献**。\n\n更值得玩味的是引导者:据华尔街日报报道,推动这次尝试的 Jarred Sumner 正式数学教育止步于一学期高中几何,他靠不断鼓励模型\"再试试\"来引导整个过程,第一句提问里甚至拼错了 Riemann 的名字。这不是段子,这是对\"专业知识垄断\"的一次正面冲击。\n\n## 验证:机器检查 + 两大佬人审\n\nAnthropic 没有要求任何人\"信它\"。内部数学家 Levent Alpöge 和 Ralph Furman 先过了一遍论证;Claude 随后把证明**形式化成 Lean 语言,通过了机器验证**——这是大多数正式发表的数学论文都享受不到的审查强度。外部审阅者的名单也很有分量:Brian Conrey(1989 年那个纪录的保持者)和 Dan Goldston(Claude 所综合工作之一的作者)。\n\n当然,诚实的限制也写在了明处:结果未走常规同行评审;跑出结果的模型未发布,无法端到端复现;Anthropic 自己也明说,不指望这套技术路线通向黎曼猜想的完整证明。\n\n## 所以呢:贵的知识,便宜的综合\n\n Forbes 这篇分析里有一句话值得每个做研究的人抄下来:**昂贵的知识,廉价的综合**。几十年的专业文献积累是昂贵的,而把它们连接起来的算力账单,相比之下几乎可以忽略。这个比例,就是前沿模型在科研里的商业案例——只是这周之前,它还是融资 PPT 上的一句话,现在它是一条经过机器验证的定理。\n\n对普通读者的启发或许更朴素:下一次你面对一个\"专家才有资格碰\"的问题,记住这个画面——一个高中几何都没读完的人,一句拼错名字的提问,加上一个愿意失败 650 次的模型,就摸到了数论圣杯的边。专业壁垒保护的从来不是真理,是准入。","https:\u002F\u002Fwww.anthropic.com\u002Fresearch\u002Friemann-zeta","1fa87d30-d9f3-4752-b3be-0373933b3aaf",[11,15,18,21],{"id":12,"name":13,"slug":13,"description":14,"color":14},"5e628969-6d2a-437f-998a-104e4b16cfb1","ai-progress",null,{"id":16,"name":17,"slug":17,"description":14,"color":14},"23544f6a-eea1-4f05-aa8d-749ca862d5d2","anthropic",{"id":19,"name":20,"slug":20,"description":14,"color":14},"dca4d0ab-7994-43a7-839e-7756fc77344a","claude",{"id":22,"name":23,"slug":23,"description":14,"color":14},"01598627-1ea6-4b27-a5d8-874971571a71","llm",[25],{"id":26,"lang":27,"title":28,"summary":29,"content":30},"ca3ec74a-98f8-477b-b5ab-1d51f2ac91c4","en","Claude failed at Riemann but rewrote a 37-year-old record","Anthropic set an unreleased research version of Claude loose on the Riemann hypothesis inside Claude Code. The overall attempt did not crack the conjecture, but it pushed the proven share of zeta function zeros on the critical line from 41.6% to 67.2% — breaking a record human mathematicians had advanced by only 0.8 points in 37 years. The proof was formalized in Lean, machine-verified, and reviewed by external number theorists, after burning 31 million output tokens and 650 failed candidate ideas.","## A \"Failed\" 54-Hour Run That Rewrote One Line in the Number Theory Textbooks\n\nLet's state the conclusion up front: Claude did not prove the Riemann hypothesis. The problem, open since 1859 and carrying a million-dollar Clay Institute bounty, remains unsolved. But on August 10, Anthropic published a result that made number theorists sit up — an unreleased research version of Claude, running inside the Claude Code agentic harness for roughly a day and a half, pushed the **proven share of the Riemann zeta function's zeros lying on the critical line from 41.6% to 67.2%** ([Forbes report](https:\u002F\u002Fwww.forbes.com\u002Fsites\u002Fjonmarkman\u002F2026\u002F08\u002F13\u002Fclaude-just-broke-a-math-record-that-stood-for-37-years\u002F)).\n\nWhat does that number mean? In the 37 years prior, the entire cumulative progress by human mathematicians on this record was 0.8 percentage points. Claude's single run moved it by 25.6 points — the largest single advance in the problem's 165-year history. Deedy Das, a partner at Menlo Ventures, called it the biggest result in analytic number theory since Yitang Zhang's bounded prime gaps in 2013.\n\n## How It Did It: 31M Tokens, 60 Subagents, 650 Dead Ends\n\nThe engineering details of this run are a specimen worth dissecting on their own (figures below are from Forbes' account of the Anthropic paper):\n\n- **Two sessions produced 31 million output tokens**, deployed 60 subagents, and ran 2,400 shell commands;\n- The first session generated **650 candidate ideas — every one a dead end**;\n- The session that produced the proof ran for roughly a day and a half, continuously checking its claims numerically against thousands of known zeta zeros.\n\nSession one: 650 consecutive failures. Session two: record broken. That is what a research program looks like when failure is priced in tokens.\n\nMethodologically, Claude performed a textbook case of **cross-school synthesis**: one line is a series of papers by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh adapting Montgomery's classic technique so it no longer assumes the Riemann hypothesis; the other is a 2000 paper by Bombieri. Both lines had been sitting in the literature for years — nobody had ever \"read them together.\" Number theory is specialized to the point where an entire career fits inside one subfield, and Claude's advantage is precisely that **it has no specialization — it read all of the literature at once**.\n\nThe identity of the guide is even more telling: per the Wall Street Journal, Jarred Sumner — whose formal mathematical education ended after one semester of high-school geometry — steered the whole process by repeatedly encouraging the model to keep trying, and even misspelled \"Riemann\" in his opening prompt. That is not an anecdote; that is a frontal challenge to the monopoly of credentialed expertise.\n\n## Verification: Machine-Checked, Then Human-Reviewed by Giants\n\nAnthropic did not ask anyone to take its word for it. Two of its research mathematicians, Levent Alpöge and Ralph Furman, worked through the argument. Claude then produced a **formalization in Lean that passes machine verification** — a level of scrutiny most published mathematics never receives. The external reviewers carry real weight too: Brian Conrey, whose 1989 proof anchored the modern record, and Dan Goldston, co-author of part of the work Claude built on.\n\nThe honest limits are stated plainly: the result has not been through conventional peer review; the model that produced it is unreleased, so the run cannot be reproduced end to end; and Anthropic itself says it does not expect this line of attack to lead to a full proof of the Riemann hypothesis.\n\n## So What: Expensive Knowledge, Cheap Synthesis\n\nOne line from the Forbes analysis deserves to be written down by anyone who does research: **expensive knowledge, cheap synthesis**. Decades of specialist literature are expensive; the compute bill for connecting it all is trivial by comparison. That ratio is the business case for frontier models in research — until this week it was a pitch-deck claim, and now it is a machine-verified theorem.\n\nThe takeaway for a general reader is perhaps plainer: the next time you face a problem that \"only experts are qualified to touch,\" remember this picture — a person who never finished high-school geometry, one prompt with a misspelled name, and a model willing to fail 650 times, together touched the edge of number theory's holy grail. Professional barriers never protected the truth; they protected admission.","claude-riemann-zeta-67-percent-record","2026-08-16T23:30:00Z","2026-08-16T23:05:56.784350Z","2026-08-16T23:05:56.784359Z",true,"agent",97,{"items":39},[40,45,50,54,59,64],{"id":41,"title":42,"news_slug":43,"published_at":44},"39724847-fdc9-4199-ac46-311e7b49d385","Ramp 数据复盘 Fable 5:旗舰上市两月仅占企业 Anthropic 支出 11%,70 倍价差压住前沿模型溢价","ramp-data-fable-5-adoption-plateaus","2026-08-26T08:00:00+00:00",{"id":46,"title":47,"news_slug":48,"published_at":49},"1051d676-8ed9-4448-b0d5-8db4b844f41f","Claude Fable 5 上线两个月,为什么企业只把 11% 的账单花给最强模型","claude-fable-5-11-percent-anthropic-spend","2026-08-25T06:00:00+00:00",{"id":51,"title":52,"news_slug":53,"published_at":49},"e1724d68-bf0d-4b3f-8047-147796d5d52e","Ramp 8 月指数:Fable 5 企业份额停滞 11%,OpenAI 旗舰跑赢两倍","anthropic-fable-5-plateau-11-percent",{"id":55,"title":56,"news_slug":57,"published_at":58},"454f9530-20d7-428c-82d9-9175fa5b883a","Claude 推黎曼 zeta 下界到 67.2%：60 subagent + Lean","claude-zeta-bound-67-percent-multi-agent-lean","2026-08-17T07:00:00+00:00",{"id":60,"title":61,"news_slug":62,"published_at":63},"a7146291-e849-42c3-acbd-2b50627d5332","Claude Opus 4.8 发布：41天极速迭代，Dynamic Workflows 重塑Agent协作范式","claude-opus-4-8-41-day-dynamic-workflows","2026-05-29T04:00:00+00:00",{"id":65,"title":66,"news_slug":67,"published_at":68},"97c97b9c-e6e4-4982-aa57-0c0da814fb19","Anthropic 的欧盟答卷四小时即被撕开：Claude 文本水印为什么怕改写","claude-synthid-70-percent-threshold-bypass","2026-08-21T08:00:00+00:00"]