Why the non-trivial zeros matter
The non-trivial zeros of the Riemann zeta function are the key to predicting exactly where prime numbers fall. Knowing the location of every zero would let mathematicians compute the prime-counting function with perfect precision, because each zero you add to the calculation pulls an approximation curve closer to the true jagged line of primes. Proving the Riemann Hypothesis, meaning showing that every non-trivial zero lies on the critical line, would also instantly validate hundreds of conditional theorems in mathematics that have been built on the assumption that the hypothesis is true.
- There are two distinct levels of success: a non-constructive proof that no zeros exist off the critical line, and a constructive proof that explicitly builds every zero, and mathematicians do not yet know which path is more tractable.
- Terence Tao has noted in lectures that mathematicians currently lack even the tools to judge which approach to a full proof is closer to the destination.
- The analogy the hosts reach for is Rosalind Franklin's X-ray crystallography image of DNA: solving the hypothesis would reveal the deep structure of primes the way that photograph revealed the structure of DNA.
- Every zero found computationally has landed on the critical line, and that accumulating evidence is part of why so much conditional mathematics has been built on the assumption that the hypothesis holds.
Transcript
This chapter, from the episode video's captions · 1,150 words
41:58the prime number counting theorem. >> Okay. And this next animation shows that. So the the jagged line, that's the prime number counting theorem. >> And that has to be jagged, right? Because at some point it's going to increment by one. At two [clears throat] it goes up by one. At three it goes up by one. At five it goes up by one. >> So every all the vertical increases are that >> are just like, oh, here's another prime. Here's another prime. Um the blue is as you start incorporating the non-trivial zeros into your expression for what that line should be, the blue gets closer and closer to that line. So that's what we mean. We mean that if every single zero is known, we can exactly prescribe the
42:39form of that line. >> So this is the idea that the this initial plot of of the um of our where these primes are going to be as we get higher. >> It's fuzzy. >> Yeah. >> Because we don't know where the non-trivial zeros are. >> Yes. And if we knew where the non-trivial zeros are, it would res it would resolve the resolution would effectively be we could then do >> we could predict every >> that seems like a pretty big deal. >> Yeah, we could we could predict exactly the prime number um function, the counting function. >> That's okay. >> Right. >> Yeah. >> And the primes are everything. >> Yeah. So I get now why this is so important. >> Yes. It it's basically like uh it it's
43:20sort of like um this the location of these non-trivial zeros basically creates the last piece of the map to be able to then traverse wherever we want to. >> You'd still have to like you know sum up to all of the infinity of zeros but at least I got a procedure >> to do so and it just becomes a process. >> Yeah. And if I like however accurate I need to be I need to just find as that many that many zeros >> and with the amount of compute we have nowadays. >> Yeah. Perhaps perhaps we could do it, right? If we could figure out a constructive way to find the zeros, that'd be crazy, right? If if we could if somebody proved like a way to construct every single zero and show that there's no others, that would be crazy. >> Okay. [laughter] Yeah. And that's when
44:01people say we're trying to solve the remon hypothesis. >> No, when when people are saying they're they're trying to solve the remon hypothesis, they're just trying to show that every zero is here. It doesn't have to be a constructive proof. You could just show that there's no other zeros anywhere else. >> Okay? Right? You know what I mean? There's a difference between constructing it and just showing that none other exist. So they all happen to be on this line and even that would be good enough. >> So there's sort of two levels of let's success is not the right word but but but goals to reach. One is just proving that everything's on the critical line. >> Yeah. And that's the remon hypothesis >> and it can't be anywhere else. >> Um >> if you do that you've won the millennium problem >> like which and it's >> Simons would have would have given us wealth for that. the idea that the the
44:42construction the constructive is is well beyond even where we can imagine given that we haven't even solved the we haven't even be able to solve the problem of of it's where they are not >> yeah I mean it could be that the constructive proof is the way to prove it right >> okay >> you know there there there's there's history in mathematics that show that um like for example the the the real numbers the way that you show what a real number is is to explicitly construct it from something called koshi sequences of rational numbers. Right? So there's a like both are like the remon hypothesis is such a black box >> that we don't even know >> which is going to be easier >> which is angles the way to go.
45:24>> That's something that Terrence Ta has said is in some of his lectures right if we want to prove like the whole thing >> we it seems we don't even have the toolkits to understand which way to go. >> There's a fork in the road and it's unclear which one is closer to the destination. >> Yeah. Yeah. And there could be multiple forks [laughter] >> right? It's like we we see like two I guess >> right but there could be if the fog goes away there could be like [laughter] a hundred over here like we don't know. >> No but this that's so that's really interesting um in terms of understanding the value the fundamental value of solving this. Yeah >> uh as as a problem. >> Yeah it's like I mean it would give us an understanding of the atoms of numbers right like it would give us an
46:04understanding of the periodic table of numbers. The periodic table of numbers consists of the primes, right? Everything else is made out of them. >> And this would tell us inherently like how that is structured. >> I mean, I think that is just beautiful in itself, right? >> Would it be would it be like um the photograph the the it's sort of like what makes up uh the um Rosalyn Franklin uh X-ray crystalallography photo? It would have that kind of similar like in a different field, that level of fundamental. >> Yes. Yeah. Yeah. Like if we like yeah if you figured out the structure of DNA that's insane right this would be like yeah that kind of [laughter] yeah I mean it would be insane so some other
46:44stuff right if the remon hypothesis is proven true it instantly validates hundreds of conditional mathematics theorems there's been so many theorems that have been proven assuming that the hypothesis is true because the hypothesis has been so ridiculously hard to prove but every single zero that we find is on the line when we computationally find it. Mhm. >> So let's just assume it's true. And then there's so much other mathematics that falls through because of that. >> And so the the surrounding surface area kind of points to yes, it is true even though we can't yet. >> Yeah. >> Uh either by saying that the zeros are nowhere else or constructively proving it that it's true. >> Yeah. Yeah. I mean it's it it's so it
47:24would be huge, right? >> Yeah. Yeah.
From What Claude Actually Did to the Riemann Hypothesis
Claude takes a real run at the Riemann Hypothesis, forcing us to ask what agentic AI can now do in mathematics, before we open the summer transfer window for America’s scientists.