Riemann–Roch for the ring $\mathbb{Z}$ - Connes, Alain; Consani, Caterina (Comptes Rendus. Mathématique)
We show that by working over the absolute base $\mathbb{S}$ (the categorical version of the sphere spectrum) instead of ${\mathbb{S}[\pm 1]}$ improves our previous Riemann–Roch formula for ${\overline{\operatorname{Spec}\mathbb{Z}}}$. The formula equates the (integer valued) Euler characteristic of an Arakelov divisor with the sum of the degree of the divisor (using logarithms with base 2) and the number $1$, thus confirming the understanding of the ring $\mathbb{Z}$ as a ring of polynomials in one variable over the absolute base $\mathbb{S}$, namely $\mathbb{S}[X], 1+1=X+X^2$.