A century of progress toward the critical line
The Riemann hypothesis asks whether all non-trivial zeros of the Riemann zeta function lie on the critical line at real part one-half. For over a century, mathematicians have been trying to prove what fraction of those zeros actually sit on that line. G.H. Hardy showed in 1914 that infinitely many zeros lie on the critical line, but that still left open the possibility that they represent zero percent of all zeros. Atle Selberg proved in 1942 that the fraction is greater than zero, Levenson pushed it to 33 percent in 1974, Conrey reached 40 percent in 1989, and by 2022 the best-known bound stood at 41.7 percent. In 2026, Claude brought that figure to 67.25 percent.
- The distinction between 'infinitely many on the critical line' and 'a positive proportion on the critical line' matters because one infinite set can still be zero percent of a larger infinite set, a subtlety the hosts work through explicitly.
- Selberg won the Fields Medal in 1950, and his positive-proportion result was considered a landmark even though his proof left the actual percentage unspecified.
- Levenson's 1974 method involved multipliers that effectively blur the Riemann zeta function and work with that smoothed version, though the hosts note they do not fully understand the mechanism.
- Hadamard and de la Vallee Poussin used Riemann's complex-analytic methods to prove the prime number theorem roughly 40 years after Riemann's original paper.
Transcript
This chapter, from the episode video's captions · 1,250 words
47:28>> You kind of get it now, right? >> That's a big deal. >> Yeah. So this is why everyone's like reload isn't anything until it really sol well if it solves it then that anyway. >> Yeah. If it solves it then that'd be great. It hasn't solved it and we're going to get into exactly what it did. Okay. So >> um after Remon's paper it took about 40 years to really iron out the proof of the prime number >> counting theorem and to show that like it followed. Um, Reman did a lot of the leg work, but then there was Jacus Hatamard, the guy behind Hatamard Matrices for those in quantum computing. And um, also I don't know how do you say this? Can you can you try? >> Charles Jean Dea Valet Poin.
48:10>> Yeah, there there we go. So th those two guys, they they proved um using Remon's complex analytic methods. And in 1914, the great English mathematician um GH Hardy, who is um shown here, he's most well known for um discovering Rammanujan. >> Mhm. >> But he's also well known for some of the math that he did. >> One of the things that he did was show that there are an infinite number of zeros on the critical line. >> Okay. Before we didn't even know that. We just knew that they were in the strip. Reman said [clears throat] they're in that strip. Hardy said there's an infinite number in that line. Now, that says a lot. It says that there's an infinite number of zeros. >> Mhm. >> It also says almost nothing about the
48:53Remon hypothesis because it could be that there's many many infinitely more elsewhere on that strip. Right. >> Right. So, it's it's a step in the right direction [laughter] >> I'd say, >> but it's not quite the whole thing. Right. [snorts] >> Um, >> next we've got finally in in 1942 at Lelay Selberg. >> Mhm. he makes a step in the direction of the remon hypothesis. Okay. >> Um he won the Fields Medal later in 1950 and this is a photo of him at the Institute of Advanced Study at Princeton. So he proves that there is a pro there is a positive proportion of
49:34non-trivial zeros on the critical line. What that means is there is some percent of zeros at least at least some percent of non of of zeros on that 1/2 line. >> So the point is there there are these there's this number of non-trivial zeros. >> There's infinitely many >> there's infinitely many because of uh Hardy. >> Mhm. >> And some percentage of those cuz we don't yet know if they're only on the critical line. >> Yeah. Yeah. And so if they're only on the critical line it would be 100%. Right. >> He said the percentage is more than zero. >> Zero. Right. >> I don't know what the number is. >> Right. >> I'm just telling you that the percentage is more than zero. >> There's at least one.
50:15>> Yeah. >> On the critical line. >> Well, no, we know that there this is this is I should I should be be careful here. >> We've already proven that there's an infinite number, >> right? >> On the critical line, right? But here's what I mean by it could be 0%. There could be an infinite number on the critical line and then there could be for every single zero an infinite number elsewhere on the strip. So for every zero there's an infinity elsewhere. And so even though I have an infinite number, it's still 0% of the total infinity. >> Total infinities because in the larger search area there's an equal amount of there's similarities. >> This is the stuff I was talking about. >> Infinity everywhere. >> Yeah. [laughter] So so you know what I mean. >> No, that's a that's a good distinction. No, that that's a good distinction. It's kind of weird to cuz we're dealing with
50:56some weird stuff here. So, we're dealing with infinities, but that infinity >> could still be 0%. He showed it's not. It has to be more than zero. >> Okay. >> Okay. >> Which is okay. We're we're from zero, we went to non zero. Okay. Everyone was super excited. He won the Fields Medal. Okay. He's like, "All right, that's dope." All right. [laughter] Um, so next we have Levenson in 1974. He pushes that number up to 33%. >> Okay, that seems like a big deal, right? So he's like a third >> of the zeros at least. >> At least a third of the non-trivial zero >> zeros are on the critical line >> are on the critical line. >> Infinitely many 34%. >> Yeah. Yeah. It's like it's like for it's a crude way of saying it is for every
51:38for every non-trivial zero on the critical line there are at most two >> outside. Right? Like I could make like a pairing and be like for this one I'm going to take two. For this one I'm going to take two. And if you do that you'll cover everything. >> Understood. that that's how that's how you want to think about it. Yep. Okay. So even though there's an infinity of it, you can still make a case that there's a third of that infinity is on the critical line and the the rest is elsewhere. So that's what Levenson shows. Okay. >> Mhm. >> Um and he does this weird thing with like mullifiers where he basically tries to like blur out the reman zeta function and then work with that blurred out version. I don't know how it works, but that's that's what Wikipedia said. [laughter] Okay. So So and and now we're
52:18getting into stuff that I really don't understand. Right. No. Fair. Fair. >> Um, so, so now I'm going to be regurgitating what I sort of figured out with my readings. Um, full disclosure, >> Levenson puts it up to 1/3. >> Um, other people take his method and start inching that up. >> So, we've got Conre in 1989, he pushes it up to two- fifths. >> Mhm. >> So, 40%. >> This is now uh uh what, 15 years later? >> Yeah. 15 years later. Um, at in 2022, so this is what 30 years later. >> Yeah. Yeah. We're up to 41.7%. >> So over Yeah. So So there was a quick jump uh by 33 to to to There's a quick There's an immediate jump to 30 3 from 0
53:00to 34. >> Yeah. There's like three big jumps. I'd say you go from zero to not zero. >> Right. Yes. Yes. >> Which even though on the graph it's like >> at the bottom >> the fact that it starts >> right. >> Right. >> Yes. >> That's huge. >> Sberg. Yeah. >> And 42. >> 42. Um and then in 74 it goes up to 34 [clears throat] per 7. big jump. >> Then it goes to 40.9. Another pretty big jump. And then it took us 30 years to get to 41.7. >> Just to even get one more%. >> Yeah. >> Not even not even one more. >> Okay. And now Claude comes in in 2026 and it has brought it up to 67.25%. >> Mhm. >> Okay.
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.