Following the earlier proof by Claude (#1544977, with the prompter famously just contributing by saying things such as "you need to believe in yourself"), published here as: https://arxiv.org/pdf/2608.13637, humans have now proposed a newer and cleaner way of achieving the same result: https://arxiv.org/pdf/2609.02882
Abstract. We obtain a new, conceptually simpler, unconditional proof that more than 67.25% of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and that at least 83.62% of the non-trivial zeros are distinct. A proof of these results was very recently produced by an internal research version of Claude developed by Anthropic and subsequently verified by Alpöge and Furman. This argument is technically intricate, and its main mechanism is not immediately transparent. It combines several ingredients from linear algebra, including a finite-dimensional matrix representation of Weil’s Hermitian form and a rank-trace inequality for Hermitian matrices, with a second moment calculation over the zeros using the explicit formula. Our new proof is shorter and proceeds by replacing the entire finite-dimensional matrix framework by a Hilbert space inequality, which allows for a direct application of Montgomery’s theorem on the pair correlation of zeros of the zeta function, in the unconditional form obtained by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.
Apparently, that's the big problem with AI proofs in math: LLMs are unable to judge what is trivial and what is complicated, thus making AI proofs very hard to understand.
I don't know how much the AI proof helped the authors come up with this new proof, but this follows well from the new role humans need to adopt to survive in math according to Terence Tao (#1551859 and #1551655). AI is not yet able to make consistently simple proofs that will make their way into textbooks. If it does not enter textbooks (or at least, the corpus of math understood by humans), it remains a useless contribution to math.
Fascinating times.
As a little anecdote, I asked my LLM last week whether a certain experimental result could be understood from symmetry arguments, and it came up with a whole mathematical framework on why it has to be so. Yet, I'm still struggling to understand the proof, as it is convoluted and uses math I am not familiar with. Yet, intuitively, I think there should be a simpler way that avoids using that specific kind of math.
My first thought was haha I guess all my contributions are useless, but then I realized that some of my work actually would be in textbooks of certain subjects (should one be written) and I felt a flash of satisfaction
Did you use Claude? I definitely think Claude is skirting very close to the line of "throw up enough complicated stuff to sound smart and make it hard for someone to check whether you're right"
Even in the way it writes, it loves to use heavy jargon and lots of numbers in a way almost designed to make it hard to follow without a pencil and paper
Used ChatGPT. I subscribed to Claude again this week, so I will see if it comes up with a similar argument as ChatGPT.
In a similar vein, I think the number of citations is a good proxy to decide whether a certain piece of research was worth one's time (and society's taxes). When I referee a paper, I always know quite well whether the article will get cited a lot or will only end up with some self-citations by the authors before disappearing into oblivion. Unfortunately, the majority of publications these days fall in the latter category. One can hope that some positive side effects remain, such as the training of the author to improve his critical thinking skills, but I'm skeptical.