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#riemannzetafunction#analyticcontinuation#riemannhypothesis

But what is the Riemann zeta function? Visualizing analytic continuation

3Blue1Brown

8.7M subscribers • 42 videos on Hallway

YouTube
  • Recommended by 6 people: 6 on Hacker News.
  • A classic. People kept recommending it from 2017 to 2023.

Intermediate. Assumes working knowledge.

5.3M views9 years ago6 recommendations

Computer scienceMath for programmers

3Blue1Brown provides a visual explanation of the Riemann zeta function, analytic continuation, and the Riemann hypothesis.

Unraveling the enigmatic function behind the Riemann hypothesis Help fund future projects: patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 3b1b.co/zeta-thanks Home page: 3blue1brown.com/ Posters/shirts for this visualization at 3b1b.co/store Music by Vince Rubinetti: soundcloud.com/vincerubinetti/riemann-zeta-function Check out some of Vince's other work here: vincentrubinetti.com/ For those who want to learn more about complex exponentiation, here are a few resources: - My video on the topic: youtu.be/mvmuCPvRoWQ - Mathologer's: youtu.be/-dhHrg-KbJ0 - Better Explained: goo.gl/z28x2R For those who want to learn more about the relationship between 1+2+3+4+... and -1/12, I'm quite fond of this blog post by Terry Tao: goo.gl/XRzyTJ Also, in a different video "What does it feel like to invent math", I give a completely different example of how adding up growing positive numbers can meaningfully give a negative number, so long as you loosen your understanding of what distance should mean for numbers: youtu.be/XFDM1ip5HdU Interestingly, that vertical line where the convergent portion of the function appears to abruptly stop corresponds to numbers whose real part is Euler's constant, ~0.577. For those who know what this is, it's kind of fun to puzzle about why this is the case. Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: goo.gl/WmnCQZ Various social media stuffs: Website: 3blue1brown.com Twitter: twitter.com/3Blue1Brown Patreon: patreon.com/3blue1brown Facebook: facebook.com/3blue1brown Reddit: reddit.com/r/3Blue1Brown

Why it's here

6 people recommend this

What people say. Highly recommended for its intuitive visual explanations of complex math and the Riemann zeta function.

lauriloveon Hacker News • Recommends

On “An Interesting Pattern in the Prime Numbers: Parallax Compression”

Appropriate juncture to repost this: link to 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.
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st1x7on Hacker News • Recommends

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 - link to 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.
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vqon Hacker News • Recommends

On “The Riemann Hypothesis, explained”

3Blue1Brown recently did a video on the Riemann zeta function: link to this video It's not as in depth but it has some helpful visualisations that I've never seen elsewhere.
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denzil_correaon Hacker News • Recommends

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. link to this video
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slavik81on Hacker News • Recommends

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: link to this video
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jstx1on Hacker News • Recommends

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 - link to 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…
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jerfon Hacker News • Mentions

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…
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fn-moteon Hacker News • Mentions

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: link to this video
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