Residues of a tropical zeta function for convex domains
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913057928642560 |
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| author | Kalinin, Nikita Lupercio, Ernesto Shkolnikov, Mikhail |
| author_facet | Kalinin, Nikita Lupercio, Ernesto Shkolnikov, Mikhail |
| contents | 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 $t\rightarrow 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21709 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Residues of a tropical zeta function for convex domains Kalinin, Nikita Lupercio, Ernesto Shkolnikov, Mikhail Number Theory Mathematical Physics Algebraic Geometry Symplectic Geometry 11M41, 14T90, 53A15, 11P21, 14M25 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 $t\rightarrow 0$. |
| title | Residues of a tropical zeta function for convex domains |
| topic | Number Theory Mathematical Physics Algebraic Geometry Symplectic Geometry 11M41, 14T90, 53A15, 11P21, 14M25 |
| url | https://arxiv.org/abs/2604.21709 |