Legendre duality for certain summations over the Farey pairs

Abstract

Each irreducible fraction $p/q>0$ corresponds to a primitive vector $(p,q)\in\mathbb Z^2$ with positive coordinates. Such a vector $(p,q)$ can be uniquely written as the sum of two primitive vectors $(a,b),(c,d)\in\mathbb Z_{\geq 0}^2$ spanning a parallelogram of oriented area one. We present new summation formulas over the set of such parallelograms. These formulas depend explicitly on $a,b,c,d$ and thus define a summation over primitive vectors $(p,q)=(a+c,b+d)$ indirectly. Equivalently, these sums may be interpreted as running over Farey pairs, i.e. pairs of fractions $0\leq c/d 2/3$ and diverges at $s=2/3$.