But what is the Riemann zeta function? Visualizing analytic continuation

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Strong rec laurilove · 2018-05 · on “An Interesting Pattern in the Prime Numbers: Parallax Compression”

Appropriate juncture to repost this: [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.

Strong rec st1x7 · 2021-01 · on “How I learned to love and fear the Riemann Hypothesis”

I don't think it comes close to 3blue1brown's video on the same topic - [this 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.

Recommends vq · 2017-01 · on “The Riemann Hypothesis, explained”

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

Recommends denzil_correa · 2018-06 · on “Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis”

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

Recommends slavik81 · 2021-01 · on “How I learned to love and fear the Riemann Hypothesis”

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: [this video]

Recommends jstx1 · 2023-09 · on “A Duodecade of Numberphile”

> 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 - [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

Cites jerf · 2022-02 · on “Mathematical outreach: The good, bad und ugly”

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 nu

Cites fn-mote · 2023-10 · on “Echoes of Electromagnetism Found in Number Theory”

> 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]: [this video]