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Interactive video lesson plan for: Visualizing the Riemann zeta function and analytic continuation

Activity overview:

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":

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There are posters for this visualization of the zeta function at

Music by Vince Rubinetti:

Check out some of Vince's other work here:

For those who want to learn more about complex exponentiation, here are a few resources:
- My video on the topic:
- Mathologer's:
- Better Explained:

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:

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:

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.


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:

Various social media stuffs:

Tagged under: Riemann,Riemann hypothesis,zeta function,mathematics,complex analysis,Riemann zeta function,blue,analytic continuation,brown,math,3b1b

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