I recently commented for this article in Gizmodo on Atiyah’s supposed proof of the Riemann Hypothesis. In the article I assert there is an internal inconsistency (in fact there are several) in his paper. This post is meant to direct people to the issue I had in mind.

Atiyah claims to construct a “weakly analytic” auxiliary function T. I do not claim to understand his construction. That said, he lists many properties of T; enough to see that no such function exists. Many commenters have focused on the claim that T is locally polynomial — this implies that it is globally polynomial, which is contradicted by, for example, property 2.5 of the paper in question. However, because Atiyah never defines the notion of “weak analyticity,” it is hard to be sure if this objection is valid.

That said, here are my tweets on a simpler issue, which I think cannot be dealt with by any reasonable relaxation of what it means to be a function:

It seems to me this contradiction is not really resolvable.