  We prove a Riemann-Roch theorem of an entirely novel nature for divisors on
the Arakelov compactification of the algebraic spectrum of the integers. This
result relies on the introduction of three key concepts: the cohomologies
(attached to a divisor), their integer dimension, and Serre duality. These
notions directly extend their classical counterparts for function fields. The
Riemann-Roch formula equates the (integer valued) Euler characteristic of a
divisor with a slight modification of the traditional expression in terms of
the sum of the degree of the divisor and the logarithm of 2. Both the
definitions of the cohomologies and of their dimensions rely on a universal
arithmetic theory over the sphere spectrum that we had previously introduced
using Segal's Gamma rings. By adopting this new perspective we can parallel
Weil's adelic proof of the Riemann-Roch formula for function fields including
the use of Pontryagin duality.
