Visualizing the Riemann hypothesis and analytic continuation





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Published on Dec 9, 2016

Understanding the Riemann hypothesis requires understanding a certain function which is famously confusing outside its "domain of convergence", but a certain visualization sheds light on how it extends.

There are posters for this visualization of the zeta function at http://3b1b.co/store

Thank you to everyone supporting on Patreon: https://www.patreon.com/3blue1brown

Music by Vince Rubinetti: https://soundcloud.com/vincerubinetti...

Check out some of Vince's other work here: http://www.vincentrubinetti.com/

For those who want to learn more about complex exponentiation, here are a few resources:
- My video on the topic: http://youtu.be/mvmuCPvRoWQ
- Mathologer's: https://youtu.be/-dhHrg-KbJ0
- Better Explained: https://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: https://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: https://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.

Special shout-outs to the following Patreon supporters: CrypticSwarm, Ali Yahya, Dave Nicponski, Damion Kistler, Juan Batiz-Benet, Yu Jun, Othman Alikhan, Markus Persson, Joseph John Cox, Luc Ritchie, Shimin Kuang, Einar Wikheim Johansen, Rish Kundalia, Achille Brighton, Kirk Werklund, Ripta Pasay, Felipe Diniz, dim85, Chris , Michael Rabadi, Alexander Juda, Mads Elvheim, Joseph Cutler, Curtis Mitchell, Ari Royce, Bright , Myles Buckley, Andy Petsch, Otavio Good, Karthik T, Steve Muench, Viesulas Sliupas, Steffen Persch, Brendan Shah, Andrew Mcnab, Matt Parlmer, Naoki Orai, Dan Davison, Jose Oscar Mur-Miranda, aidan boneham, Henry Reich, Sean Bibby, Paul Constantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Jeffrey Herman, Jacob Young and Steve Muench.


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