Residues of a tropical zeta function for convex domains

Abstract

We define an $\operatorname{SL}_n(\mathbb{Z})$-invariant tropical zeta function of a convex domain. In dimension 2 it admits boundary Dirichlet-series representation with summands indexed by Farey pairs. For $C^3$ strictly convex domains, it extends meromorphically to $\Re(s)>3/5$, holomorphic there except for a simple pole at $s=2/3$, with residue proportional to equiaffine perimeter. A Tauberian argument yields the $t^{1/3}$ wave-front lattice-perimeter asymptotic for $t\rightarrow 0$.