Residues of a tropical zeta function for convex domains

Abstract

We define an ${\mathrm{SL}}_n(\mathbb{Z})$-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 universally proportional to equiaffine perimeter. A Tauberian argument yields the $t^{1/3}$ wave-front lattice-perimeter asymptotic for $t\to 0$. In addition, for a special domain $L$, which is a limit shape of lattice polygons in a square, with its tropical zeta function being expressed in terms of Witten SU(3) zeta function, we compute the exact coefficient in the asymptotic expansion of the integer-averaged lattice point counting for the leading term $N^{1/2}$.

116 pages, 13 figures and schematic diagrams. Main theorem: for smooth strictly convex planar domains, the tropical zeta function continues meromorphically to Re(s)>3/5 with simple pole at s=2/3; residue gives equiaffine perimeter. Includes Tauberian asymptotic for lattice perimeter of tropical wave fronts, and a detailed demonstration of a novel scheme for lattice point counting