



The Riemann zeta function is given by this simple formula when the complex number s has Re(s) > 1. Then the sum converges! But we can "analytically continue" the Riemann zeta function to define it for other values of s, and that's where the fun starts.



(continued)

The Riemann zeta function is zero for some numbers with 0 < Re(s) < 1. These are called the "nontrivial zeros" of his zeta function. Riemann computed a few and hypothesized they all have Re(s) = 1/2.



https://www.youtube.com/watch?v=sD0NjbwqlYw …

Riemann found a formula for the number of primes < n as a sum over the nontrivial zeros of the zeta function. My first tweet shows the sum over the first k nontrivial zeros.



So, if the Riemann Hypothesis is true, we'll get a better understanding of primes!



(continued)

So far people have checked, using a computer, that the first 10,000,000,000,000 nontrivial zeros of the Riemann zeta function have Re(s) = 1/2.



This might seem like damned good evidence for the Riemann Hypothesis.



But maybe not!



(continued)

For me, the best evidence for the Riemann Hypothesis is that it's part of a much bigger story! Mathematicians like Weil, Grothendieck and Deligne proved similar results for related functions.



Much remains mysterious, though.



(continued)



https://www.youtube.com/watch?v=sD0NjbwqlYw …

I bet that Atiyah's claimed proof, if and when he writes it up, will not convince experts.



In 2017 he claimed to have a 12-page proof of the Feit-Thompson theorem, which usually takes 255 pages:



https://www.maths.ed.ac.uk/~v1ranick/atiyahtimes2017.pdf …



He showed it to experts, and... silence.



(continued)

In 2016 Atiyah put a paper on the arXiv claiming to have solved a famous problem in differential geometry. The argument was full of big holes:



https://mathoverflow.net/questions/263301/what-is-the-current-understanding-regarding-complex-structures-on-the-6-sphere …



So, I'm not holding my breath this time.



But of course I'd be happy to be wrong.



(the end)





Here is Atiyah's lecture: on the Riemann Hypothesis



https://www.youtube.com/watch?v=UBVy0oOYczQ …



Here, apparently, is his paper:



https://www.dropbox.com/s/pydoj0a8hguebc6/2018-The_Riemann_Hypothesis.pdf?dl=0 …



It refers extensively to this much longer paper, where he attempts to compute the fine structure constant:



https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view …

Here's what Science says about Atiyah. Only a few mathematicians were willing to be quoted, and I drew the short straw.



By the way, I have huge respect for Atiyah, whose earlier work revolutionized geometry and physics.



http://www.sciencemag.org/news/2018/09/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis …

You can follow @johncarlosbaez.

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