A hat genus

Notation: \(\widehat{ \operatorname{A}}_g (M)\). Defined as the multiplicative sequence genus of the power series \begin{align*} \frac{\sqrt{z} / 2}{\sinh (\sqrt{z} / 2)}=1-\frac{z}{24}+\frac{7 z^{2}}{5760}-\cdots \end{align*}
Genus: like a ring homomorphism \(g: {\mathsf{K}}_0(X, \coprod, \times) \to ({\mathbf{Z}}, +, \cdot)\) for \(X\) a manifold with boundary up to cobordism, where \(g(X) = 0 \iff X = {{\partial}}X'\).
In \({\mathbf{Z}}\) for spin manifolds, and is even if additionally \(\dim_{\mathbf{R}}M = 4 \operatorname{mod}8\)
The Aatiyah-Singer Index Theorem implies that \(\widehat{ \operatorname{A}}_g (M) = {\mathsf{Ind}}(\mkern-3mu \not{ \partial} )\) for spin manifolds