Links to Atiyah’s preprints on the Riemann hypothesis

Sir Michael Atiyah’s preprints are now on the internet:

  1. The Riemann Hypothesis
  2. The Fine Structure Constant

The meat of the claimed proof of the Riemann hypothesis is in Atiyah’s construction of the Todd map $T:\C\to \C$. It supposedly comes from the composition of two different isomorphisms
$$\C\xrightarrow{t_+} C(A)\xrightarrow{t^{-1}_{-}} \C$$ of the complex field $\C$ with the $C(A)$, the center of a von Neumann hyperfinite factor $A$ of type II-1. Understanding Atiyah’s work boils down to understanding this Todd map, and therefore in understanding what is in the paper “The Fine Structure Constant”.

Assuming there is a zero $b$ of the Riemann zeta function $\zeta(s)$ off the critical line, Atiyah defines a function
$$F(s) = T(1 +\zeta(s + b)) – 1.$$ The function $F$ satisfies $F(0) = 0$ because one of the properties of the Todd map $T$ is that $T(1) = 1$, and $F$ is also supposedly analytic. According to some of the basic properties of the Todd function which is a polynomial on a closed rectangle containing this zero, this would imply that $F(s) = 0$ and therefore that $\zeta$ is identically zero, which is the contradiction.

Now, I have very little understanding of von Neumann algebras so I won’t comment at all on the Todd map. I have no doubt that the experts will dissect this because there’s so much attention on it. Even assuming all the properties of the Todd function, I find the proof difficult to follow. For example, the assumed zero $b$ off the critical strip: I can’t find where “$b$ is off the critical strip” is even being used. In fact, it’s hard to see where any of the basic properties of the zeta function are being used.

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