Update again, Sept 2021: Just found this excellent link to R. J. Lipton taking Atiyah seriously. I don't have time to dig in right now, but want to. I'm delighted to see someone doing this, as the binary response to Atiyah's insights left me feeling like more people should be looking into this, just in the hope that someone with Atiyah's solid reputation and skills might provide some clues in the legendary pursuit, even if that's all he provided. What's most exciting about this post I just found is that I'm currently studying quaternions, and I'm still curious about the link between Clifford Algebra and octonions and the very intriguing comment by Furey that she intuits a different way of doing division, which aligns with my intuitive approach to solving this...
Update: This is an accurate snapshot of how I explore mathematics, as someone who hated math and had nothing to do with it for the first 35 years of my life. No one credible currently believes Atiyah's proof which I'm endorsing below, but, as this is all over my head, and as I navigate by intuition not rigor, I have fun moments like the following one. I have learned to let such moments happen, trusting future events will sort out whether the truth was discovered, or just another phantom. It turns out that Atiyah and Furey had communicated with each other, so the coincidence in similarity was not random, as I thought at the time. The following is the post as I wrote it on September 25, 2018:
Oh Lordy I'm having a blast digging into Atiyah's proof of Riemann's Hypothesis. I'm so delighted to find the link between Atiyah's insights and Furey's fascinating discoveries which I also looked into with great interest, both on the days they were published.
Here are two mathematicians talking about the same underlying things, both of them leaving everyone around them baffled and surprised at what looks to be... on the surface... one of the greatest discoveries in math AND physics ever.
Cohl Furey has discovered "The Peculiar Math That Could Underlie the Laws of Nature" and Michael Atiyah has discovered "A Proof of Riemann's Hypothesis" (pdf) which is so profound it not only crowns math but goes on further and bridges math and physics.
I feel certain they are right, even if only because the beauty pouring out of this alignment between math and physics is outta this world. Atiyah has even included prime numbers and the fine structure constant! (pdf) I had no idea they were related, but I can see the connection -- the fine structure constant is built from multiplication in a way that is similar to how primes are composed by addition (and... you heard it here first, upon a foundation of division underneath all that addition).
More soon. I'm posting stuff on this in my social media account which I should be posting here... You may have noticed I'm completely disregarding all the people who are doubting Atiyah. Not me! I'm in! I've been circling around these same themes for years, and am impressed by what I'm seeing here even though much of it is over my head.
[Very lightly edited, for example, I've included a link to a future post. What a fun day that was. Pretty much all mathematicians are unanimous that Atiyah did not discover the proof he claimed. Of course I secretly still think Atiyah discovered the proof and all other mathematicians missed what he really meant.]