Residues of a tropical zeta function for convex domains

TL;DR

This paper introduces an $ ext{SL}_n( ext{Z})$-invariant tropical zeta function for convex domains, analyzing its properties and asymptotics, especially in dimension 2 and for smooth convex shapes.

Contribution

It defines a new tropical zeta function invariant for convex domains and explores its meromorphic extension, residues, and asymptotic behavior.

Findings

In dimension 2, the zeta function has a boundary Dirichlet-series representation.

For $C^3$ strictly convex domains, it extends meromorphically with a simple pole at $s=2/3$.

The residue at the pole is proportional to the equiaffine perimeter.

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 $t\rightarrow 0$.