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Following a theorem of David R. Hayes, we give a geometric interpretation ofthe special value at $s=0$ of certain $1$-cocycle on$\mathrm{PGL}_2(\mathbb{Q})$ previously introduced by the author. This workyields three main results: an explicit formula for our cocycle at $s=0$, ageneralization and a new proof of Hayes' theorem, and an elegant summationformula for the $0$th coefficient of the Ehrhart quasi-polynomial of certaintriangles in $\mathbb{R}^2$.

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