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Numberphile v. Math: the truth about 1+2+3+...=-1/12

Confused 1+2+3+…=-1/12 comments originating from that infamous Numberphile video keep flooding the comment sections of my and other math YouTubers videos. And so I think it’s time to have another serious go at setting the record straight by having a really close look at the bizarre calculation at the center of the Numberphile video, to state clearly what is wrong with it, how to fix it, and how to reconnect it to the genuine math that the Numberphile professors had in mind originally.

This is my second attempt at doing this topic justice. This video is partly in response to feedback that I got on my first video. What a lot of you were interested in were more details about the analytic continuation business and the strange Numberphile/Ramanujan calculations. Responding to these requests, in this video I am taking a very different approach from the first video and really go all out and don't hold back in any respect. The result is a video that is a crazy 41.44 (almost 42 :) minutes long.

Lots of amazing maths to look forward to: non-standard summation methods for divergent series, the eta function a very well-behaved sister of the zeta function, the gist of analytic continuation in simple words, etc.

The original Numberphile video is here
https://youtu.be/w-I6XTVZXww . Also check out the links to further related Numberphile videos and write-ups in the description of that video.

Here is a link to Ramanujan’s notebook that contains his Numberphile-like 1+2+3+… = -1/12 calculation. http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook1/chapterVIII/page3.htm
This notebook entry was also one of the starting points of my last video on this topic: https://youtu.be/jcKRGpMiVTw

Other good videos that deal with this strange “identity” include the following:
https://youtu.be/0Oazb7IWzbA (a Numberphile video featuring the mathematician Edward Frenkel who is also talking about the connection between the Riemann Zeta function and Ramanujan's crazy identity.)
https://youtu.be/sD0NjbwqlYw (a nice 3Blue1Brown video about visualizing the analytic continuation of the Riemann Zeta function).

If you know some calculus and want to read up on all this, beyond what is readily available via the relevant Wiki pages and other internet resources, I recommend you read the last chapter of the book by Konrad Knopp, Theory and applications of infinite series, Dover books, 1990 (actually if you know German, read the extended version of this chapter in the 1924 (2nd) edition of the book "Theorie und Anwendung der unendlichen Reihen". The Dover book is a translation of the 4th German edition. The 5th German edition from 1964 can be found here: https://gdz.sub.uni-goettingen.de/id/PPN378970429).

People usually recommend Hardy's book, Divergent series, but I'd say only look at this after you've looked at Knopp's book which I find a lot more accessible. Having said that, Hardy's book does have quite a bit of detail on how Ramanujan summation applies to the Zeta function; see chapters 13.10. and 13.17.

The article by Terry Tao that I mentioned at the end of the video lives here: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

Thank you very much to my mathematician friend Marty Ross for all his feedback on the script of this video and for being the grumpy voice in the background and Danil Dmitriev the official Mathologer translator for Russian for his subtitles.

Enjoy :)

Burkard

P.S.: Here is a scan of the page from that String theory book that is shown in the Numberphile video. Note, in particular, the use of equal signs and arrows on this page. http://www.qedcat.com/misc/String_theory_book.jpg

For today's maths t-shirts google: "zombie addition math t-shirt", "label your axes math t-shirt".

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