Riemann takes the zeta function into the complex plane
The Riemann zeta function extends beyond summing 1/n² and similar series by allowing the exponent to be a complex number, such as 2+i. In the complex plane, each term in the sum produces a rotation rather than a step along the number line, so the partial sums trace a spiral that converges to a single point whenever the real part of the exponent is greater than one. Below that threshold, the series diverges. The grid of input values with real part greater than one maps neatly onto roughly half the complex output plane, which immediately raises the question of how to define the function for the rest of the plane.
- The hosts credit 3Blue1Brown's YouTube video on the Riemann zeta function for the animations used to illustrate how inputs map to outputs in the complex plane.
- The specific example worked through is zeta(2+i), whose output the hosts give as approximately 1.15 minus 0.44i.
- When the exponent is a real number like 2, every term lands on the number line and the sum converges to π²/6, the answer to the Basel problem; the imaginary part of the exponent is what introduces the rotation.
- The term 1 to any complex power always equals 1, so every spiral starts at the same point before the subsequent terms begin curving the path.
Transcript
This chapter, from the episode video's captions · 906 words
25:55remon said no what if we took one over stuff to the power of a complex number so it's like 1 + i or 2 plus i. What what would that do? And it turns out there's a very well- definfined way of talking about that. So here's how it works with complex numbers. Okay, here this I'm going to borrow a lot of um these visuals from three blue one brown and a lot of our viewers are going to recognize it as such because he has an absolutely amazing video on YouTube about the remon zeta function. Um and he's an abs he's the goat of mathematical visuals. So um >> big shout out >> big shout out. So here's here's what happens if I take remon zeta to the 2 plus i.
26:36>> So in this case we're the primary thing we're changing is uh the exponent. Yes. In the denominator. >> Yeah. Because that's how the remon zeta this function is defined. >> Right. The function is defined as the sum over one over the natural numbers to some power. You choose the power. >> Yeah. Right. And so we're just saying we've now chose the power. >> Yeah. We chose we choose the power to be 2 plus i instead of two. If it was just two, it would be the basil problem again. and then I'd get square. But because it's 2 + i, if it's just 2, then every single term I'm just moving along the number line, right? Uh for 1, I'd get 1 + 1/4 + 1 9th and I'd go a little bit little bit closer to p<unk> square over 6 and I'd just be moving on the
27:17number line. With complex exponents, now I'm rotating on the complex plane. The complex plane is defined as the real numbers on the x-axis, the imaginary numbers on the y-axis. So that's why 2 plus i is two on the x, one on the y. 2 + 2 i would mean two on the x, two on the y. So that's that's how complex numbers are defined. And the way that you do an exponent is you actually start rotating where that's happening. Now 1 to the power of anything is just one. So that's why you go one first. But then two, 1 / 2 to the^ of 2 plus i. That's going to curve a little bit downward. And it's going to curve a little bit downward for for the three. And it's going to spiral to that point. Mhm. >> So the remon zeta function takes as
27:58input 2 plus i >> and maps it to that point down there >> which is like 1.15 minus 0.44 i. Okay, that's how it works in the complex plane. >> Yes. >> Now turns out as long as the real part is greater than one, your sum is going to be fine. And we've got an animation here courtesy of three blue one brown. Again, >> as I change the input, the spiral is going to change, but it's going to it's going to be well defined. It's going to end up somewhere. >> Okay. So, here I'm taking my input, which is the yellow dot, and the output of the remon zeta function is where that spiral ends up. Okay. Perfectly well-
28:39definfined function. [clears throat] Really nice. Okay. >> Mhm. >> Um, let's see what this function does to the grid lines. Okay. Here we've seen individual points getting mapped. Now, let's see what this function does to the grid lines. as long as the power the real part is greater than one. This is how the the the grid lines >> Mhm. >> map. >> Okay. Now, why do we need the powers to be greater than one for the real? That's because, you know, imagine if I did um the the power equals zero, right? Then I just have 1 + 1 + 1 + 1 that's going to diverge to infinity. [clears throat] >> Does it's not well defined at least in the series representation. But as if my powers are greater than one, then each
29:20fraction gets smaller and smaller and so my spiral actually converges somewhere. That's the idea. >> Okay, got it. >> Okay. Um and so this is what's happening. All of the all of the points on the right are getting mapped to these points that are in sort of half of the plane. >> But now this kind of begs the question, what about the other half? >> Right. >> Right. It seems like I could just draw some lines, [laughter] >> right? I could just continue the function right? >> Right. Like I could I could make up >> how the rest of the function would behave given I know so well how this function behaves over here. >> We're and we're and specifically you're
30:01referencing to the to the left of of that line of the y ais. >> Yeah. Yeah. Exactly. There's a little part on the on the right of the y- axis too that that thing is not Right. It's like about like half. >> It's a little there. >> Yeah. But like but like my point is >> the parts where we can use the series. >> Yes. >> Give me this really nice functional map. >> Right. >> Okay.
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