Riemann zeta function

In compute, it will have a direct application there.

The Riemann zeta function results, number theory, and cryptographic research.

There's no immediate practical cryptographic breakthrough that follows, but it does open up a new path of discovery for cryptographic study that is gonna be really important, obviously.

One of the things, by the way, just let me pause there.

Just to make that point, the Riemann hypothesis is arguably the most important problem in number theory.

It states that the Riemann zeta function, which concerns the distribution of prime numbers, has zeros only at even integers and complex numbers with a real component of one-half.

Instead, it's easier to gauge how big a moment this is by observing how professional mathematicians reacted.

Alex Kontorovich, a Rutgers professor, wrote, "Quasi-Riemann hypothesis? Are you kidding me? If a human did this, it would be an instant Fields Medal, no questions asked." Indeed, most of the initial takes came from mathematicians that have been relatively positive about AI's contribution to the field.

We've made some kind of incremental progress towards the Riemann hypothesis.

It- it's just a, a famous mathematical problem that currently has a million dollars attached to its head, that it, it says something about the zeros of a certain function, the Riemann zeta function.

As of yesterday night, we now know a slightly bigger region where there are definitely no zeros of this function.

So in some sense, it's incremental progress towards the grand hypothesis, but it's certainly not a proof of the hypothesis.

I mean, humans have proved thousands, tens of thousands of theorems, and AI has proved hundreds, right? So, you know, AI is still way behind.

Right.

So you, you enable these layers of metacognition, which I think are incredible, right? So telling the student, "Hey, you're the kind of student that really needs the challenge, and I'm going to now take a side tour, and I'm gonna tell you about the Millennium Prize, you know, in mathematics and the Riemann zeta hypothesis.

And though we're not gonna solve that because it hasn't been solved yet, you know, those are the kinds of things that are gonna really inspire you." And for somebody else, it might be something else.

And, and actually providing those, those insights to the student at the same time that you're actually leveraging them to create a better experience is like, this world is gonna be amazing.

They were using AI to find it.

And this other gentleman that we'd mentioned, uh, Levant Elpoge, works at Anthropic, and he was involved with another mathematical discovery from August on a set of problems called the Riemann zeta problem.

And what they did for the Riemann zeta problem is they took a one piece of established research from the last, uh, couple of years, which uses this very, a thing called the narrow box conditions to move the Riemann zeta problem forward, and based on their prompting, the LLM found some research from the '70s on how to remove narrow box conditions from other similar problems, and their LLM- Combine that together and came up with a proof on this other thing.

The difference here is that Levon is consistently working with other mathematicians.

He's consistently talking about things, and keep this number in mind.

It concerns the distribution of prime numbers and carries a $1 million millennium prize.

Claude did not prove it, no one expected it to, but during its attempt, it improved a related lower bound, the fraction of zeros of the Riemann zeta function that aligns with the hypothesis.

It pushed that bound from 41.6% to 67.2%.

The result was reviewed by Anthropic's own mathematicians and then shared with outside experts.

a French mathematician who made major contributions to number theory, complex analysis, differential geometry, and partial differential equations.

He obtained his doctorate in 1892 and was awarded the Grand Prix des Sciences Mathematiques for an essay that he wrote on the Riemann zeta function.

Talked about that before.

Increasingly, he was caught up in the Dreyfus affair because his second cousin, Laurie, was married to Dreyfus.