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

Following a theorem of Hayes, we give a geometric interpretation of the special value at $s=0$ of certain $1$-cocycle on ${\mathrm{\text{PGL}}}_2(\mathbb{Q})$ 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 zeroth coefficient of the Ehrhart quasi-polynomial of certain triangles in ${ℝ}^2$.