Tropical formulae for summation over a part of SL(2, Z)

Abstract

Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c{)}^2+(b+d{)}^2}$, let $(a,b,c,d)$ stand for $a,b,c,d\in {\mathbb{Z}}_{\ge 0}$ such that $ad-bc=1$. Define \begin{equation} \label{eq_main} F(s) = \sum_{(a,b,c,d)} f(a,b,c,d)^s. \end{equation} 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 $F(s)$ converges when $s>1$ and diverges at $s=1/2$. (This papers differs from its published version: Fedor Petrov showed us how to easily prove that $F(s)$ converges for $s>2/3$ and diverges for $s\le 2/3$, see below.) We also prove $$\sum _{\begin{array}{c}(a,b,c,d),1\le a\le b,1\le c\le d\end{array}}\frac{1}{(a+b{)}^2(c+d{)}^2(a+b+c+d{)}^2}=1/24,$$ 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.

typos corrected