Today the internet has been getting excited about Sir Michael Atiyah’s claimed proof of the Riemann Hypothesis, which he presented at the Heidelberg Laureate Forum this morning. We’ve collected all the relevant links and tweets to help you make sense of what’s going down in critical-line-town.

Firstly, Aperiodical roving reporters Katie and Paul were present in the room, allowing @Aperiodical to live-tweet a blow-by-blow of the talk itself. The thread is here (click the date-time link to see the whole thread):

Our @stecks is in the room with Atiyah at #hlf2018, but can't tweet and listen. We'll try to relay what's going on.

— The Aperiodical (@aperiodical) September 24, 2018

The talk itself was live-streamed by the conference, but due to heavy traffic their streaming servers, and the backup servers they switched to, were unable to support the load. One plucky reporter streamed it on Periscope, but the actual video has now been put up on the conference’s YouTube channel:

The actual paper Atiyah’s presenting has been doing the rounds online, and there are links to a 5-page PDF outlining the proof sketch and agreeing with the content of the talk, as well as a paper on the Fine Structure Constant referenced in that one which has more detail.

Some immediate analysis in this Twitter thread by Markus Pössel, who was also in the room (click the date-time link to see the whole thread):

https://twitter.com/mpoessel/status/1044131977950109696

Markus has also now written a brief post for the HLF blog about what will happen next.

New Scientist has posted a follow-up to its initial announcement, with a more cautious treatment of what has happened, including quotes from friend of the site Nicholas Jackson.

Here are a few non-rigorously-selected discussions of the mathematics from around the web (again, click the timestamp in tweets to see the full thread):

https://twitter.com/fvaccarino/status/1044006102596276224

https://twitter.com/chocolitt88/status/1044025759671742464

A Reddit thread attempting to collect discussion in the wake of the lecture:

Atiyah’s lecture on the Riemann Hypothesis

byu/AngelTC inmath

There’s also a lively Twitter discussion on the reaction to the proof, in this thread:

I see this attitude all over math Twitter that it will be embarrassing for Atiyah if his proof of the #riemannhypothesis doesn't pan out. As if aiming for a big prize and missing is a bad thing.

— John D. Cook (@JohnDCook) September 24, 2018

And finally, this MathOverflow thread What is the definition of the function T used in Atiyah’s attempted proof of the Riemann Hypothesis? provides some insight into the mysterious Todd function.

If any further developments occur, we’ll post about it as soon as we hear. The Heidelberg Laureate Forum continues for the rest of the week, and Paul and Katie will attempt to write about any other interesting maths as it happens, here and over on the HLF blog.

It is not aimimng for a grand prize and missing that is wrong. It is the premier mathematician not being vigilant enough to guard against hasty or wishfull attempts that degrade his reputation and paint him in colours worthy of cranks. I have an admiration for Atiyah, the man, and his work and I feel sorry that he let himslef be looked upon as a spoiler of his prestigious past achievments.

This proof doesn’t work (at least for me). Assuming the Todd function T(s) has a few of the given properties and b is the first critical zero going up the critical line, referring to Artiyah’s preprint we can supposedly assume T(0)=0, T(1)=1, T is a set isomorphism of the complex numbers (I have that from (3.4) of his preprint on the fine structure constant given at ICM2018 in Rio), and for all power series f(s) with no constant term (converging on a closed and bounded subset say) we have T(1+2f(s))=T(1+2f(s)+f(s)^2), so f(s)^2=0, hence f(s)=0. This looks troubling.

Indeed, for f(s)=T(1+zeta(s+b))-1 we get f(s)=0 as before so T(1+zeta(s+b))-1=0, so T(1+zeta(s+b))=1=T(1), thus zeta(s+b)=0 and zeta(s) everywhere. Note we have not used anything about zeta which one doesn’t accept. Thus no function T(s) with the 4 properties exists.

Yup