The Barnes-Hurwitz zeta cocycle at $s=0$ and Ehrhart quasi-polynomials of triangles

Following a theorem of David R. Hayes, we give a geometric interpretation of the special value at $s=0$ of certain $1$-cocycle on ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathbb{Q})$ previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at $s=0$, a generalization and a new proof of Hayes’ theorem, and an elegant summation formula for the $0$th coefficient of the Ehrhart quasi-polynomial of certain triangles in ${\mathbb{R}}^2$.