Tropical formulae for summation over a part of

Abstract

Let , let stand for $a,b,c,d\in \mathbb Z_{\geqslant 0}$ such that $ad-bc=1$. Define

In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one.We prove that converges when $s>1$ and diverges at $s=1/2$. We also prove that

$$\begin{aligned} \sum \limits _{(a,b,c,d)} \frac{1}{(a+c)^2(b+d)^2(a+b+c+d)^2} = \frac{1}{3}, \end{aligned}$$

and show a general method to obtain such formulae. The method comes from the consideration of the tropical analogue of the caustic curves, whose moduli give a complete set of continuous invariants on the space of convex domains.