ChatGPT Prompts

Useful AI and tech tools 👇 https://shir-man.com/

- Claude solved the Riemann hypothesis. - Claude won the million-dollar prize. - The Riemann hypothesis is now proven. - AI can solve all math problems. - The result replaces human mathematicians. It does mean: - Claude apparently made a meaningful contribution to a hard area of mathematics. - AI can sometimes combine existing human knowledge in new ways. - AI may be useful as a research assistant or even a research collaborator. - Progress in AI mathematical reasoning is continuing. --- ## The main message The article is saying: > Anthropic asked Claude to attempt the Riemann hypothesis. Claude failed to prove it, but unexpectedly proved a strong related result: it raised the known guaranteed percentage of zeta-function zeros on the critical line from 41.6% to 67.2%. This is not the full Riemann hypothesis, but it is an example of AI making real progress in advanced mathematics.

It generated many ideas, used many subagents, ran commands, wrote Python scripts, checked numerical examples, reviewed proofs, searched existing papers, and re-proved the result independently. Some key points: - Claude first tried 650 ideas, but none worked. - Then it tried again with many “subagents” — basically separate Claude processes working on different parts. - It ran thousands of numerical checks. - It downloaded many math papers to see whether the result was already known. - It wrote a paper. - It helped create a formal proof in Lean. - Human mathematicians reviewed the result. The article also says the human mostly encouraged it with messages like “keep going” and “believe in yourself.” That does not mean Claude has feelings; it just means the prompts caused it to continue exploring rather than giving up too early. --- ## What does “lower bound” mean? A lower bound is a minimum guaranteed amount. If someone says: > “At least 67.2% of the zeros are on the line.” that is a lower bound. It does not say exactly how many zeros are on the line. It only says: > “No matter what, we can guarantee at least this many.” So the old lower bound was 41.6%. The new lower bound is 67.2%. --- ## What are “zeros of the Riemann zeta function”? A “zero” of a function is an input where the function equals zero. For example, if: > f(x) = x − 3 then f(3) = 0, so x = 3 is a zero. The Riemann zeta function is much more complicated and uses complex numbers. Its zeros contain information about prime numbers. The Riemann hypothesis is about whether all the important zeros lie on one specific line. --- ## More precise version If you want the mathematical version: Let ζ(s) be the Riemann zeta function. The nontrivial zeros are zeros inside the “critical strip,” where the real part of s is between 0 and 1. The Riemann hypothesis says: > Every nontrivial zero has real part equal to 1/2. The result discussed in the article is about the proportion of nontrivial zeros whose real part is 1/2. Roughly speaking, if you count zeros up to some height and take a limiting proportion, the article says Claude proved: > The limiting lower bound for the proportion of zeros on the critical line is at least 67.2%. Previously, the best known bound mentioned in the article was 41.6%. --- ## What is the technical idea, in simple words? The article gives a very technical description involving quadratic forms, Weil, positive-definite and negative-definite subspaces, moments, Hilbert transforms, and so on. In simple words, the idea seems to be: 1. Mathematicians already had powerful tools for studying zeros. 2. Recent work made some of those tools usable without assuming the Riemann hypothesis is true. 3. Claude combined several existing results in a new way. 4. It looked at the whole mathematical structure together, instead of breaking it into simpler separate parts. 5. That allowed it to get a stronger bound on how many zeros must be on the line. The article says the “courage” was to treat the full space all at once, including both zeros on the line and possible zeros off the line. Here “courage” is a human-like word. It means Claude tried a more ambitious approach that may have been difficult or unpopular among humans because it is complicated. --- ## Is the result definitely true? According to the article, the proof was checked in several ways: - Anthropic mathematicians reviewed it. - External experts examined it. - Claude produced a formal Lean proof. - The formal proof passed a validation tool. But in mathematics, new results usually still need broader community review. So the safest way to say it is: > Anthropic claims Claude produced a valid proof of the improved bound, and experts have examined it, but the wider mathematical community may still review and validate it further. --- ## What this does not mean It does not mean:

## Short version The page is an Anthropic blog post saying: > An unreleased research version of Claude tried to solve the famous Riemann hypothesis. It did not solve it. But while trying, it proved a related mathematical result: it improved a known lower bound for how many zeros of the Riemann zeta function lie on the “critical line” from 41.6% to 67.2%. In other words: Claude did not prove the Riemann hypothesis, but it apparently made a real advance on a related problem. --- ## What is the Riemann hypothesis? The Riemann hypothesis is one of the most famous unsolved problems in mathematics. It is about the Riemann zeta function, a complicated mathematical function that is deeply connected to the distribution of prime numbers: 2, 3, 5, 7, 11, 13, etc. The zeta function has special inputs where its value becomes zero. These are called zeros. The important ones are the nontrivial zeros. The Riemann hypothesis says: > All nontrivial zeros of the Riemann zeta function lie on one special vertical line in the complex plane, called the critical line. So the hypothesis is basically: > “All the important zeros are exactly on this line.” Nobody has been able to prove this since it was proposed in 1859. There is also a famous million-dollar prize for solving it. --- ## What did Claude actually do? Claude did not prove that all the zeros are on the line. Instead, it proved something weaker but still important: > At least 67.2% of the important zeros are on the line. Before this, mathematicians had been able to prove that at least 41.6% of the zeros are on the line. So the improvement is: | Result | Guaranteed proportion of zeros on the line | |---|---:| | Previous known bound | 41.6% | | Claude’s claimed result | 67.2% | That means: mathematicians already knew that at least about 41.6 out of every 100 zeros must be on the line. According to the article, Claude showed that at least about 67.2 out of every 100 zeros must be on the line. But 100% would be needed to prove the Riemann hypothesis. So Claude did not solve the main problem. It improved a related guarantee. --- ## A simple analogy Imagine you have a huge field full of trees. Someone claims: > “Every tree in this field is an oak tree.” That is like the Riemann hypothesis: all zeros are on the line. You cannot prove that yet. But you can prove something smaller: > “At least 41.6% of the trees are oak trees.” That was the old mathematical result. Now Claude supposedly proved: > “At least 67.2% of the trees are oak trees.” That is progress. But it still does not prove that all the trees are oak trees. --- ## Why is this considered important? Because the Riemann hypothesis is extremely hard. Even small improvements in understanding the zeros of the zeta function are considered meaningful in number theory. The zeros are connected to how prime numbers are distributed. If we understand the zeros better, we understand primes better. Also, going from 41.6% to 67.2% is a large jump in this area. The article says this result was checked by mathematicians at Anthropic and also examined by outside experts, including Brian Conrey and Dan Goldston, who are known experts in this field. It also says Claude produced a formally verifiable proof using Lean. That means the proof was translated into a machine-checkable format, so a computer system can verify that the logical steps are valid. --- ## Did Claude solve the Riemann hypothesis? No. The article explicitly says Claude did not succeed in proving the Riemann hypothesis. It says Claude “took a real stab” at it, meaning it made a serious attempt. But the main problem remains unsolved. The result is described as an “unintended byproduct” of trying to solve the bigger problem. --- ## How did Claude do it, according to the article? The article says Claude used a research version of Claude Code and worked over two long sessions.