Abstract

An asymptotic formula is presented for the number of planar lattice convex polygonal lines joining the origin to a distant point of the diagonal. The formula involves the non-trivial zeros of the zeta function and leads to a necessary and sufficient condition for the Riemann Hypothesis to hold.

Highlights

  • In 1979, Arnold [1] considered the question of the number of equivalence classes of convex lattice polygons having a prescribed area A

  • Vershik changed the constraint in this problem and raised the question of the number, and typical shape, of convex lattice polygons included in a large box [−n, n]2

  • That the limit shape of a typical convex polygonal chain is the arc of parabola tangent to the sides of the square, which maximizes the affine perimeter

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Summary

A statistical mechanical model

We start this paper by reminding the correspondence between lattice convex chains and non negative integer-valued functions on the set of pairs of coprime positive integers. Let P be the set of primitive vectors, that is to say, the set of all vectors (x, y) whose coordinates are coprime positive integers, including the pairs (0, 1) and (1, 0). As already used in Jarník [8], the space of lattice increasing convex chains starting from the origin is in one-to-one correspondence with the space Ω of nonnegative integer-valued functions ω : P → Z+ with finite support (that is to say, ω(v) = 0 only for finitely many v ∈ P):. The function ω associated to the convex chain (xi, yi)0≤i≤k is defined for all v ∈ P by ω(v) = gcd(xi+1 − xi, yi+1 − yi) if there exists i ∈ {0, . The endpoint of the chain is equal to ω(v) v

Description of Sinai’s model and overall strategy
Integral representation of the partition function
Analysis of the partition function
First theorem
Second theorem
Numerical considerations
Digitally convex polyominoes
A Analytic continuation of the Barnes function
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