What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

Question

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the definition of $T(s)$?

Atiyah states that this function is defined in his paper "The Fine Structure Constant", but I can't seem to find a copy of the paper. So can anyone tell me how Atiyah defined $T$ in that paper?

Answer 1

Here's a public paper of the "the fine structure constant" by Atiyah.

It doesn't seem to be the original, but a copy: https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view

See the section 3.4, the Todd function is defined there.

[Edit] As pointed by @T_M in the comments, here you would find the main properties of the T function.

Answer 2

In his article "The Fine Structure Constant", Atiyah says the following about T on page 6:

For the case of inverse-integer weights, Hirzebruch formalized the notion of exponential maps and found the most general solutions 1-Chapter 3. His motivation came from the behaviour of certain manifold invariants, in algebraic geometry, under taking Cartesian products. But the problem was purely formal and applied to all manifolds, giving topological invariants [especially after Atiyah-Singer index theory]. He showed that the basic example was the Todd genus, named after J.A. Todd, whose generating function, due to Bernoulli, is $\frac{x}{ 1−e^{-x}}$, already exploited by Euler. It is crucial that this function, which underpins Hirzebruch’s formalism, is analytic in the closed interval [0, 1/2] (see section 6). This ensures that the normalized trace of 4.7 extends to the weak closure A. Following Hirzebruch, we have named it after Todd and denoted it by T .

Reference 1 is the book "Topological methods in algebraic geometry (with appendices by R.L.E.Schwarzenberger, and A.Borel)" by F. Hirzebruch. Here is a link to the book.

I don't understand the definition of Todd genus in chapter 3. However, in the first section of the book it defines a multiplicative sequence $T$. Such a sequence can be viewed as a function $T:\mathbb C\times \mathbb C^{\infty}\rightarrow \mathbb C$. The definition is then: $$T(z,p_1,p_2,\dots)=\sum_{i=0}^{\infty} T_i(p_1,\dots,p_i)\cdot z^i$$ or $$T(x,c_1,c_2,\dots)=\sum_{i=0}^{\infty} T_i(c_1,\dots,c_i)\cdot x^i.$$ Here the $T_i$ are the Todd polynomials. This $T$ might be related to the $T$ that Atiyah is using.

At page 9 of the book, it states about general multiplicative sequences $K$:

In abbreviated notation we write $$K(\sum_{i=0}^{\infty} p_iz^i)=\sum_{i=0}^{\infty} K_j(p_1,\dots,p_j)z^j$$

In this notation, it seems as if $K$ is a function from $\mathbb C$ to $\mathbb C$, but that is not technically accurate. The left-hand side should actually be interpreted as $K(z,p_1,p_2,\dots)$.

This might be the reason that Atiyah presents $T$ as a function from $\mathbb C$ to $\mathbb C$. That could also be 'abbreviated notation'. Maybe, this is what actually happens:

1. We start with some $v\in \mathbb C$
2. Then write $v=\sum p_iz^i$ in some specific way
3. Now define $T(v)$ as $T(z,p_1,p_2\dots)$ for these specific values of $z,p_1,p_2,\dots$

However, I do not know how the $z,p_1,p_2,\dots$ have to be chosen. There is an infinite number of way to get the equation $v=\sum p_iz^i$.

If Atiyah was really on to something, then I think his articles contain a great lot of abbreviated notation. Therefore, most logical steps in his article are incomprehensible to us 'mere mortals'.