News regarding ABC conjecture and Riemann Hypothesis

There have been a couple news stories regarding proofs of major theorems. First, an update on Shinichi Mochizuki’s proof of the abc conjecture, then an announcement that Sir Michael Atiyah claims to have proven the Riemann hypothesis.

Shinichi Mochizuki’s proof of the abc conjecture

Quanta Magazine has a story today saying that two mathematicians have concluded that Shinichi Mochizuki’s proof of the ABC conjecture is flawed beyond repair. The story rightly refers to a “clash of Titans” because Shinichi Mochizuki and his two critics Peter Scholze and Jakob Stix are all three highly respected.

I first wrote about the abc conjecture when it came out in 2012. In find the proposed proof fascinating, not because I understand it, but because nobody understands it. The proof is 500 pages of dense, sui generis mathematics. Top mathematicians have been trying to digest the proof for six years now. Scholze and Stix believe they’ve found an irreparable hole in the proof, but Mochizuki has not conceded defeat.

Sometimes when a flaw is found in a proof, the flaw can later be fixed, as was the case with Andrew Wiles’ proof of Fermat’s last theorem. Other times the proof cannot be salvaged entirely, but interesting work comes out of it, as was the case with the failed attempts to prove FLT before Wiles. What will happen with Mochizuki’s proof of the abc conjecture if it cannot be repaired? I can imagine two outcomes.

1. Because the proof is so far out of the mainstream, there must be useful pieces of it that can be salvaged.
2. Because the proof is so far out of the mainstream, nobody will build on the work and it will be a dead end.

Michael Atiyah and the Riemann hypothesis

This morning I heard rumors that Michael Atiyah claims to have proven the Riemann hypothesis. The Heidelberg Laureate Forum twitter account confirmed that Atiyah is scheduled to announce his work at the forum on Monday.

I was a fan of Atiyah’s work when I was a grad student, and I got to talk to him at the 2013 and 2014 Heidelberg Laureate Forums. I hope that he really has a proof, but all the reaction I’ve seen has been highly skeptical. He claims to have a relatively simple proof, and long-standing conjectures rarely have simple proofs.

It would be great for the Riemann hypothesis to be settled, and it would be great for Atiyah to be the one who settles it.

Whereas Mochizuki’s proof is radically outside the mainstream, Atiyah’s proof appears to be radically inside the mainstream. He says he has a “radically new approach … based on work of von Neumann (1936), Hirzebruch (1954) and Dirac (1928).” It doesn’t seem likely that someone could prove the Riemann hypothesis using such classical mathematics. But maybe he found a new approach by using approaches that are not fashionable.

I hope that if Atiyah’s proof doesn’t hold up, at least something new comes out of it.

Update (9/23/2018): I’ve heard a rumor that Atiyah has posted a preprint to arXiv, but I don’t see it there. I did find a paper online that appeared to be his that someone posted to Dropbox.  It gives a very short proof of RH by contradiction, based on a construction using the Todd function. Apparently all the real work is in his paper The Fine Structure Constant which has been submitted to Proceedings of the Royal Society. I have not been able to find a preprint of that article.

Atiyah gave a presentation of his proof in Heidelberg this morning. From what I gather the presentation didn’t remove much mystery because what he did was show that RH follows as a corollary of his work on the Todd function that hasn’t been made public yet.

Michael #Atiyah at the #HLF18:

“Solve the #RiemannHypothesis and you’ll become famous.

But if you’re already famous, you run the risk of becoming infamous.”

Sense of humour above all!#RetransmisionesQS pic.twitter.com/SWU1gXvm87

— QuinteScience (@QuinteScience) September 24, 2018

Update: More on Atiyah’s proof here.

What are these theorems?

The abc conjecture claims a deep relationship between addition and multiplication of integers. Specifically, for every ε > 0, there are only finitely many coprime triples (a, b, c) such that a + b = c and c > rad(abc)^{1 + ε}.

Here coprime means a, b, and c share no common divisor larger than 1. Also, rad(abc) is the product of the distinct prime factors of a, b, and c.

This is a very technical theorem (conjecture?) and would have widespread consequences in number theory if true. This is sort of the opposite of Fermat’s Last Theorem: the method of proof had widespread application, but the result itself did not. With the abc conjecture, it remains to be seen whether the method of (attempted?) proof has application.

The Riemann hypothesis concerns the Riemann zeta function, a function ζ of complex values that encodes information about primes. The so-called trivial zeros of ζ are at negative integers. The Riemann hypothesis says that the rest of the zeros are all lined up vertically with real part 1/2 and varying imaginary parts.

Many theorems have been proved conditional on the Riemann hypothesis. If it is proven, a lot of other theorems would immediately follow.

Related posts

• Bernoulli numbers and Riemann zeta
• Distribution of Riemann zeta zeros
• ABC vs FLT
• My interview with Michael Atiyah