Why 1 + 2 + 3 + 4 + … = -1/12
Plugging -1 into the Riemann zeta function and applying analytic continuation gives -1/12, which is why the sum 1 + 2 + 3 + 4 + … is assigned that value. The result isn't arbitrary: Ramanujan independently reached -1/12 using a completely separate, more elementary technique, with no knowledge of complex analysis or the zeta function, which Krishna takes as evidence that -1/12 is the only defensible finite value for that series. String theorists exploit this assignment regularly to handle infinities in their calculations.
- Ramanujan mentioned the -1/12 result in his famous letter to Cambridge mathematician Hardy, warning that Hardy would probably think he belonged in an asylum.
- Hardy's reaction to the letter was surprise precisely because Ramanujan showed no awareness of complex analysis yet landed on a result that matched the zeta function approach.
- The Basel problem, summing the reciprocals of squares to get pi squared over 6, is offered as a parallel example of the same phenomenon: Fourier series, Euler's original hack, and Taylor series all converge on the same answer.
Transcript
This chapter, from the episode video's captions · 1,050 words
33:15um there was a lot of hype over the um the sum 1 plus 2 plus 3+ 4 + 5 equaling -112th. Now you can finally kind of understand why that is the case. If I plug in negative 1 to the remon zeta function, right? Then that's going to be 1 + 1 over 2. But 21 means I just flip. So I get 1 + 2 plus 1 over 31. So I just flip. Y >> right. So that's just 1 + 2 + 3 + 4. Right? It doesn't make sense on its own. But if you use the remon zeta function, you analytically continue it outside of its domain to the negative numbers and then you plug in1 to that functional thing that I showed you earlier with the
33:55signs and the gamma function and everything. You plug in negative1 to that, you're going to get112. >> And it turns out that if someone if someone were to put a gun to your head and say define this series as a number, you can't say infinity. You have to pick a number. The only correct answer is -112. Okay. To show you another reason why this is the case and why Shrinasa Rammanujan who's a big hero of mine is on this is um Rammanujan did not know anything about complex numbers very little when he was in India and he was reading like a trig book and discovering like all of math. Okay. Um he he wrote a letter to Hardy who was a
34:37big mathematician in Cambridge asking for like you know help to to go to Cambridge and like work under him. One of the things that he wrote down was this 1 plus 2 plus 3+ 4 dot dot dot equals -112. And he said, I know you'll probably think of putting me into an insane assignment. [laughter] But look, I have techniques that assign numbers to divergent series and I got -112. Hardy looked at that and he was like, hold on. This guy makes no mention of complex analysis and imaginary numbers, but he still figured out a way >> to get to -112th. And this this was a known thing at the because the Hardy is after remon right. So there you can just
35:17plug in negative 1 and you know that Rammenujan had no idea about remon about the zeta function. He figured out a really hacky way to like assign it a number which is like not exactly clean. Again hacky now we know there's reasons why it works but um he also came up with the same number112th. Just goes to show you how closed math is. it. He was able to in it from a totally separate pathway, but it still converged at the same the same destination regardless of his lack of not going down the road already traveled. >> Yeah. Yeah. He went a complete but found the same destination. It's kind of cool, right? And that's what I mean by like if someone were to put a gun to your head
35:57and say define what this is and it can't be infinity. The only correct answer is -112th. Okay. >> So fascinating. >> The real correct answer is really infinity, right? But if you were to put if you were to really put me through it, it would have to be -112th, right? And the remon zeta of negative1 is 1112th, right? It just I like I don't think it means anything to sum up the natural numbers, but like actually like string theorists use that all the time >> to like deal with their infinities. >> Yes. >> They're like a12th. It's like okay, >> the number of dimensions is the denominator. [laughter] >> They probably have some party trick where they where they do that. Um I I just thought that's that's a really cool
36:37like little aside. >> No, no, that's and that's I think it's because it's it part of what we're doing is we're connecting all of these component parts, you know, ma mathematics being almost this closed system, >> right? That has has these bounded box and and and a level of like the logic logical consistency that exists through it even if you come from different disciplines within that subject matter. Um and seeing how those different disciplines address the same question. >> Yeah. >> Um or try to approach the same questions. It's really fascinating. >> Yeah. I mean it's it's really cool. I mean the basil problem is another example of that. Let me just give you another little anecdote from my personal
37:18life. Um I I told you about math 215 at Princeton University. The first semester of math that I did. One of the problems on the final was to calculate the bezel problem. Okay. It's like great. Okay. Oiler did it. Yeah. You can do it for a final you freshman. And uh this this was Peter Sarnac. He was just a maniac. But um he actually in the problem statement he he gave us enough information and we [snorts] used Forier series >> and a clever trick with Forier series to show and I got that problem right. It was a it was a great problem. Um but yeah, you can use Forier series to show the pi squared over 6. You can use Oilers's hack. You can also use um tailaylor series. A lot of times people use like um the tailaylor series for
38:00sign and things like that. So there's so many different ways but you will always arrive at pi^ square over 6. Just the beauty of mathematics is is love. You got to love it. >> Okay. So now let's finally talk about the reman hypothesis. We [clears throat] have talked about the reman zeta function. >> Now let's talk about the hypothesis >> which is sort of a uh an the next step from starting with the basil problem >> uh that let us have a fundamental understanding of of >> we generalized it to the zeta function >> to the zeta [clears throat] function. >> Yeah. And now >> now we're finally going to get to the remon hypothesis. Okay. Now the remon
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.