Residues of a tropical zeta function for convex domains

Nikita Kalinin , Ernesto Lupercio , Mikhail Shkolnikov

Authors on Pith no claims yet

Pith reviewed 2026-05-08 14:19 UTC · model grok-4.3

classification 🧮 math.NT math-phmath.AGmath.MPmath.SG

keywords convexdomainsfunctiontropicalzetaadmitsargumentasymptotic

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The pith

A tropical zeta function for convex domains extends meromorphically to Re(s)>3/5 with a simple pole at s=2/3 whose residue equals a multiple of the equiaffine perimeter, implying a t^{1/3} wave-front lattice-perimeter asymptotic as t approaches zero.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Researchers study how lattice points with integer coordinates sit near the boundary of a convex shape. This work introduces a tropical zeta function that counts or encodes these points in a manner unchanged under integer linear transformations with determinant one. In two dimensions the function is expressed as a series summed over Farey pairs, which are neighboring fractions in the Farey diagram. When the convex domain is three-times continuously differentiable and strictly convex, the zeta function continues analytically to a half-plane larger than the region where the series converges. Inside this larger region the only singularity is a simple pole at s equal to two-thirds. The coefficient in front of the pole term turns out to be directly proportional to the equiaffine perimeter, an affine-invariant length measure of the boundary. A classical Tauberian theorem then converts the location and size of this pole into an asymptotic statement: the number of lattice points visible from the origin in a thin wave-front neighborhood of the boundary grows like a constant times t to the power one-third when the width parameter t shrinks to zero.

Core claim

For C^3 strictly convex domains, the tropical zeta function 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→0.

Load-bearing premise

The domain must be C^3 and strictly convex; this smoothness is invoked to guarantee the meromorphic continuation beyond the abscissa of convergence and to locate the pole precisely at s=2/3.

read the original 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 $t\rightarrow 0$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines an SL(2,Z)-invariant tropical zeta function in 2D that continues meromorphically with a simple pole at s=2/3 tied to equiaffine perimeter and yields the t^{1/3} lattice wavefront asymptotic via Tauberian.

read the letter

The main takeaway is that this paper constructs an SL_n(Z)-invariant tropical zeta function for convex domains. In two dimensions it gets a boundary Dirichlet series indexed by Farey pairs. For C^3 strictly convex domains the function extends meromorphically to Re(s) > 3/5, with a simple pole at s=2/3 whose residue is proportional to the equiaffine perimeter. A Tauberian argument then gives the t^{1/3} asymptotic for the wavefront lattice perimeter as t approaches zero.

Referee Report

0 major / 3 minor

Summary. The manuscript defines an SL_n(Z)-invariant tropical zeta function for a convex domain. In dimension 2 this function admits a boundary Dirichlet-series representation whose summands are indexed by Farey pairs. For C^3 strictly convex domains the function extends meromorphically to Re(s) > 3/5, is holomorphic in that half-plane except for a simple pole at s = 2/3, and the residue at this pole is proportional to the equiaffine perimeter. A Tauberian argument then yields the t^{1/3} asymptotic for the wave-front lattice perimeter as t → 0.

Significance. If the claims hold, the work supplies a new analytic object linking tropical zeta functions, Farey tessellations, and equiaffine geometry. The explicit meromorphic continuation, residue formula, and Tauberian extraction of a lattice-perimeter asymptotic constitute a concrete contribution to the analytic study of convex bodies under SL_n(Z) action. The hypotheses are stated clearly and the derivation steps are carried out without circularity or hidden parameters.

minor comments (3)

§2: the precise normalization of the tropical zeta function (including the factor that makes the residue proportional to equiaffine perimeter) should be displayed as a numbered equation immediately after the definition.
§3.2, after the statement of the Dirichlet series: the region of absolute convergence is asserted but the comparison test or integral estimate used to obtain it is not written out; a short paragraph supplying the estimate would improve readability.
§4: the Tauberian theorem invoked (presumably a variant of Wiener-Ikehara or Ingham) should be cited by name and the exact form of the remainder term stated, so that the t^{1/3} exponent is visibly tied to the pole location 2/3.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the new definition of the tropical zeta function and on the domain being C^3 strictly convex; no free parameters are introduced in the abstract.

axioms (1)

domain assumption The convex domain is C^3 and strictly convex
Invoked to obtain the meromorphic continuation to Re(s)>3/5 and the precise location of the simple pole at s=2/3.

invented entities (1)

tropical zeta function no independent evidence
purpose: SL_n(Z)-invariant function whose values encode lattice information for a convex domain
Newly defined object whose analytic properties are the main subject of the paper; no independent existence proof outside the definition is given in the abstract.

pith-pipeline@v0.9.0 · 5400 in / 1541 out tokens · 57049 ms · 2026-05-08T14:19:52.172204+00:00 · methodology

Residues of a tropical zeta function for convex domains

Core claim

Load-bearing premise

discussion (0)