May Update

Sorry for the post resurrection, but I just had to say, for those who have studied it, complex analysis is fundamentally an amazing and visually intuitive body of mathematics. If I could be so blunt I’d say it very much “feels” like the sort of 3b1b content we love haha.

Thanks for reminding us about the Patreon discount at the DFTBA store. I just snagged a Pi plushie to keep company with my Schrödinger's Cat plushie from Physics Girl! 😁

Phil Stubblefield

Oh definitely you will need to do a few videos on complex analysis, I doubt it would even be possible without it even in the 'direct' approach. Regardless, I cannot say what's the best way to do it but I can say that your channel is unique amongst the other math channels in that you try to explain things in such a way that the viewer, regardless of their background, can feel like a genius, in this case Bernhard Riemann sketching a formula for the number of primes less than a given magnitude. The videos you made on the Putnam exam and the Math Olympiad are such cases, where even a layman like me could follow through the logic, and actually sketch a proof following your explanations, even if the proof isn't rigorous. But it does make me feel like if I did sit through those test, I would have probably managed to write something down and not totally flop the test. Or even that thing you said about analytic continuation leveraging the property of the complex derivative being angle preserving. It's so obvious in hindsight, but I never thought about complex derivatives that way before, even though I noticed the Cauchy-Riemann equations reminding me of a determinant of some sort though back then I wasn't as familiar with linear algebra. I think so long as you follow such an approach, you can't go wrong. And if it requires two or three half an hour videos of complex analysis and such then go for it, because we the viewers can wait, and honestly I think the more we actually understand, the more satisfied we would be with the video or series of videos. Analogous to a story, you start with a premise, go on a journey, and end tying up all the loose ends nice and neat. Best not to try and squeeze everything by cutting out what might be small details I guess, sort of like a bad movie adaptation of a book that cuts out many small character developing plots because there's not enough time, leaving the movie feeling hollow and unsatisfying, to use the same analogy.

L'îbrarY

It's funny you should mention the zeta function because doing that follow-up is relatively high on my queue for this year. You're right that it's very difficult to do right, and I can't see a way to do it without doing a lesson on the basics of complex analysis and contour integration. That's not a problem, of course, but it means it's not a small project! It's possible to explain it more "directly" rather than referencing Fourier transforms (or really, Mellin transforms), but I'm unsure of what's necessarily the best approach. It doesn't seem too wild to me that Riemann's formula would emerge before the PNT was proved. From what I can tell, it looks less like he saw the zeta function zeros and sought a connection to primes, and it's more that the Euler product formula offered some sense that it should be possible to relate primes to something "simpler", in the form of the zeta function, and as someone steeped in complex analysis he would naturally ask about ways to poke and prod at that function. In either case, I give it a 75% chance that I end up doing that video this year.

3blue1brown

Hi, Grant ! I believe your idea of doing short videos is very interesting to reach young people that are just waiting to fall in love with mathematics. Like it or not, every content creator is competing for the attention of young people, so we must adapt ourselves to their way of consuming information to succeed

3SoME is a fantastic name and I see nothing wrong with it. Also the plushies maybe as an inside joke have a proportion of 3:1 blue to brown so that there will only ever be a ratio of 3 blue and 1 brown in existence. I would like to share some grievance with you though, since your video on the analytic continuation of the Riemann zeta was posted yeeeaaaaars ago and I'm still waiting on that continuation of the analysis (heh) on how precisely the zeta zeroes relate to the distribution of primes and specifically the number of primes less than a given magnitude (incidentally the name of Riemann's paper), but I read up on the subject and it's an insanely difficult topic to present in a simple form, but it does in fact use Fourier transforms and convolutions (in the Dirichlet series at least) so I'm guessing you might eventually build up to it hopefully within the year? Its a funny thing to me because I would have thought that one would prove the prime number theorem first BEFORE even just conjecturing an exact formula for the primes, but it is striking how Riemann conjectured the exact formula before even the PNT was solved (granted he was probably fairly sure that PNT was true), and that proving the PNT involved actually finding this exact formula first. And how did Riemann even see these random zeroes and thought "yep I can find the exact number of primes less than some number with this."? That's insane.

L'îbrarY