But what is the Riemann zeta function? Visualizing analytic continuation

3Blue1Brown recently did a video on the Riemann zeta function: [VIDEO] It's not as in depth but it has some helpful visualisations that I've never seen elsewhere.

Appropriate juncture to repost this: [VIDEO] "Visualizing the Riemann zeta function and analytic continuation" (Watch the last 30s or so if nothing else. You'll not regret it.) There is a kind of 'real' Ural spiral [or really a kind of chaotically decaying 'orbit'] related dynamic that gives the prime numbers (assuming the RH.) Conceivable that OP's pattern might have some relation to this dynamic, and fun pondering to boot.

3B1B has a nice visualization of "Riemann zeta function and analytic continuation" which might help in further understanding of this proof. [VIDEO]

I don't think it comes close to 3blue1brown's video on the same topic - [VIDEO] Still nice of the video in the original post to give a bit more of a historical context, instead of focusing entirely on the mathematics. It could have done without the building analogy around the beginning though, that didn't come up later and was just distracting.

That was a fantastic video. I was left wondering a bit about the difference between the Riemann Zeta function and the original Zeta function, but I found this 3Blue1Brown video which explains exactly that: [VIDEO]

I expect that it is because in general, more useful functions and operations are closed in the complex plane. Things that may superficially appear complicated to a human being learning math, like analytic extension [1], from a theoretical perspective means that when you need the tool, it is there, and continues to expand the sets of problems you can address. As you add restrictions to the set of numbers you want to use, you are importing those restrictions into every proof you want to do on those numbers. For instance, consider the question, "Does this polynomial of some high degree have roots, and if so, how many?" The Fundamental Theorem of Algebra proves that the answer is yes, and the number of roots is the same as the degree (although some may be roots multiple times). See [2]. Now, as you watch 2, consider what happens to the proof if you confine it even to the reals, let alone the integers. While the complex plane is "more complex" from a human perspective, the proof of the Fundamental Theorem of Algebra using the complex plane is not that complicated. Consider trying to make equivalent statements about polynomials with only real roots, and not using the complex plane in the proofs. In cases where it is feasible, your proofs will inevitably be carrying around a lot more caveats about what polynomials it applies to, and there may be things that the caveats simply render infeasible. As you step down the number hierarchy, the caveats get worse and worse. There's more and

> The 1+2+3+4+... = -1/12 video still seems a bit of a cheat, though! Because in that video they do invalid operations. Compare it to how 3blue1brown handles this - [VIDEO] - the video is on the Riemann zeta function and my timestamp is on his comment about -1/12 at the end. But trying to explain something in a nuanced way doesn't get the same views as dumbing it down and coming up with a clickbait title (not that 3blue1brown is struggling for views). I understand that the two channels have somewhat different objectves and approaches but in this case one of them strikes me as much better. The top comment under the 3blue1brown video addresses this too - you don't need to additionally mystify math.

> many people thinking mathematics is anything other than a language. I disagree with the claim that mathematics is only language, if by language you mean terms and definitions. The EASY part of mathematics is the language. There are deep truths, too, and they are not just language. For a (kind of) popular example, the question of "Where do the zeros of the Riemann zeta function lie?" is way way more than just vocabulary. There is a 20 minute video by 3Blue1Brown on the subject. [1] [1]: [VIDEO]