What is the Riemann Hypothesis?
The Riemann Hypothesis asks whether every non-trivial zero of the Riemann zeta function lies exactly on the critical line, the line at real part one-half running through the center of the critical strip. Riemann proved the non-trivial zeros fall somewhere in the strip between 0 and 1 on the real axis, then noted in his paper that they probably all sit on the center line, without proving it. The trivial zeros, by contrast, are already fully understood: they fall at all negative even numbers because a sine term in the functional form produces zeros at regular intervals.
- Riemann himself described the critical line conjecture as an interesting problem but said proving it was not necessary for his own purposes, which is why it remains a hypothesis rather than a theorem.
- The location of the non-trivial zeros is directly connected to how accurately the prime counting function n over log n describes the distribution of primes.
Transcript
This chapter, from the episode video's captions · 671 words
38:34zeta function has been used to prove the um that bound of primes the n over log n. That's actually chapter 7 of this book >> which was the red line in the previous. >> Yes, it's how do primes grow? They grow like n / log n. >> Um the distribution there and how close I get to n over log n has to do with where the zeros are for the zeta function. And what that means is what inputs map to zero if I plug in that input into the exponent [clears throat] >> of the of the you know one over the thingy and or or I the functional form. >> Yes. >> Do I get zero as the output? Okay. There
39:14are two types of zeros. There are the trivial zeros which happen at all of the negative even numbers. So if I if I put in -2 I get zero. If I put put in -1, I get that -112th. Yes. Okay. But if I put in -2, I get zero. If I put put in4, I get zero. Those are the trivial zeros. And that just has to do with the fact that there's a sign in there. And sign goes like this. >> Okay. So it's trivial. >> That's it comes back to >> Yeah. It's like the functional form has a sign in there. And so the sign because the sign oscillates, you're going to get zeros at >> at a cons at a consistent interval. >> Yeah. Yeah. And that's that's okay. Fine. >> Fine. >> Now there are the nontrivial zeros. Oh, my favorite. Non-trivial.
39:55>> Yes. And they are non-trivial for a reason because we don't know where they are. >> And that is the remon hypothesis. >> Where are they? >> Non-trivial because everything else is well defined at this point. >> Yeah. >> Except for the non-trivial zeros. >> The non-trivial zeros. We know that there's a bunch of more zeros, but we don't know where they are. Okay. Remon proved that they are inside of that strip. He said they're somewhere in here between 0 and one on the real axis. That's called the critical strip. So Remon proved that and then in his in in his paper he writes probably [laughter] they're on the critical line. They're on the one half right in the middle of that strip. >> Okay. >> Okay. That's what he writes. He's like
40:36probably >> but he's like I don't want to prove it because it it for my purposes it's not it's not necessary but it seems like an interesting problem. That's why it's called the hypothesis because he hypothesized it >> in his paper and now we're all chasing it. >> There's there's a search space that we believe these non-trivial zeros exist in. >> Yeah. >> And within that context there is a discrete line >> right in the middle. >> Right in the middle which is the sort of uh most likely place the hypothesis of within this search space >> this is where you should look. >> This is where you should look. And if we can prove that they all lie only on that line
41:16>> and nowhere else. >> They're not in [clears throat] like some fudged like part around it. Every single non-trivial zero is on that line. That is the remon hypothesis. >> We want to resolve it similarly to how cleanly the trivial zeros resolve or the -112th resolve. >> Yeah. Yeah. >> Where it's always true. >> Yes. Yes. It's always true that every single zero is going to be on that line. Okay. Okay. That that has been the quest. >> Yes. >> Okay. >> Okay. >> Oh, it seems so easy. >> Yeah. [laughter] Um and if we know why would we care, right? Well, if we know where the zeros are for the remon hypothesis, >> Mhm. >> then we could prescribe an exact form of
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.