What is the Todd's function in Atiyah's paper?

Question

In terms of purported proof of Atiyah's Riemann Hypothesis, my question is what is the Todd function that seems to be very important in the proof of Riemann's Hypothesis?

Answer 1

While there might be an interesting question here about the older math that Atiyah references, it is worth pointing out what Atiyah actually says about the function $T$:

1. $T : \mathbb{C} \to \mathbb{C}$ (this is in Section 3.4 of the longer paper "The Fine Structure Constant").

2. $T$ is "real" (see 2.2 of the shorter paper "The Riemann Hypothesis" - it doesn't mean "real-valued").

3. $T(1) = 1$ (2.3 of the shorter paper "The Riemann Hypothesis").

4. On any compact, convex set $K$, $T$ is a polynomial of some degree and the degree is in principle allowed to depend on the set $K$ (this is in the start of Section 2).

5. If $f$ and $g$ are power series with no constant term then $$ T\Bigl( 1 + f(s) + g(s) + f(s)g(s)\Bigr) = T\Bigl(1 + f(s) + g(s)\Bigr) $$ (this is 2.6 of shorter paper).

The following proof is from a Redditor : Set $f(s) = e^s - 1$ and $g(s) = 1 - e^s$. Point 5. then implies that $$ T\Bigl( 1 + e^s - 1 + 1 - e^s + (e^s - 1)(1 - e^s)\Bigr) = T(1) $$ i.e.(using 3.): $$ T\Bigl( e^s(2-e^s))\Bigr) = 1. $$ Now notice that the function $e^s(2-e^s)\rvert_{\mathbb{R}}$ takes any value in $(-\infty,1)$. To see this claim you can solve $$ e^x(2-e^x) = y\ \Leftrightarrow e^{2x} - 2e^x + y = 0 $$ by using the quadratic formula and taking logarithms to get a real solution when $y < 1$. This shows that $T\rvert_{(-\infty,1)}$ is constant.

OK so now take a compact, convex set $K \subset \{ \mathrm{Re}(z) < 1\}$ that contains an interval on the real line, i.e. $K$ contains a subinterval $I$ of $(-\infty,1)$. From the properties 2. and 4. we know that $T\rvert_K$ is a polynomial with real coefficients. But we also know it is constant on $I$ which means $T \rvert_K$ is constant.

Since $K$ was arbitrary we can easily exhaust$ \{ \mathrm{Re}(z) < 1\}$ by compact, convex sets to show that $T$ is constant on $\{ \mathrm{Re}(z) < 1\}$, in particular, this includes the critical strip.