TITLE: What is the Todd's function in Atiyah's paper?
QUESTION [45 upvotes]: 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?

REPLY [37 votes]: 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 $T$:

$T : \mathbb{C} \to \mathbb{C}$ (this is in Section 3.4 of the longer paper "The Fine Structure Constant").
$T$ is "real" (see 2.2 of the shorter paper "The Riemann Hypothesis" - it doesn't mean "real-valued").
$T(1) = 1$ (2.3 of the shorter paper "The Riemann Hypothesis").
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).
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.