Riemann zeta function – I

Visualizing the Riemann zeta function and analytic continuation

Video imported from ‘3Blue1Brown‘ YouTube channel.

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[ Publicado el 9 dic. 2016

How a certain perspective on what the Riemann zeta function looks like can motivate what it might mean beyond its domain of convergence. One month free audible trial, I recommend “The Art of Learning”:

https://www.audible.com/3blue1brown

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Support on patreon, and get early access to future “Essence of” series:

https://www.patreon.com/3blue1brown

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Music by Vince Rubinetti:

https://soundcloud.com/vincerubinetti…

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Check out some of Vince’s other work here:

http://www.vincentrubinetti.com/

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For those who want to learn more about complex exponentiation, this is actually what the first video I ever put on this channel is about (https://youtu.be/F_0yfvm0UoU). I’m not exactly proud of the pacing in that video, but it does give a less commonly shared intuition. An even better explanation is Mathologer’s (https://youtu.be/-dhHrg-KbJ0). I also think Better Explained does a good job (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

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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

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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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