
Terence Tao – Understanding 1+2+3 =-1/12 Without Complex Analysis - earthicus
https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/
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gus_massa
Tao is Tao, but I prefer to recommend the video from Mathloger
[https://www.youtube.com/watch?v=jcKRGpMiVTw](https://www.youtube.com/watch?v=jcKRGpMiVTw)
(and the followup
[https://www.youtube.com/watch?v=YuIIjLr6vUA](https://www.youtube.com/watch?v=YuIIjLr6vUA)
)

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earthicus
Tao's article certainly requires more background to read, but it gives a much
deeper insight than explanations based on Cesaro or Euler/Ramanujan summation,
which (as Tao points out in the article), in addition to being very puzzling,
are actually mutually inconsistent. His main result is that these classical
sums / the zeta function are only the _constant term_ in a certain asymptotic
expansion associated with a smoothed version of the original sum. Recognizing
the existence of the non-constant terms in that expansion corrects the
nonsensical negativeness and logical inconsistency. This is intimately
connected to the analytic continuation POV, which is discussed in section 2.
It's well worth a read if you have time over a few days. I wrote an overview
of the strategy here:

[https://www.reddit.com/r/math/comments/cx3qzv/terence_tao_un...](https://www.reddit.com/r/math/comments/cx3qzv/terence_tao_understanding_123112_without_complex/eyiun42/)

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gus_massa
The video gets to a similar point at 27:10.

