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 $0\leq c/d<a/b\leq 1$ satisfying $ad-bc=1$. The input for our formulas is the graph of a strictly concave function $g$. The terms are the areas of certain triangles formed by tangents to the graph of $g$. Several of these formulas for different $g$ yield values involving $\pi$. For $g$ a parabola we recover the classical Mordell-Tornheim series (also called the Witten series). As a nice application we also discuss formulas for continued fractions for an arbitrary real number $\alpha$ involving coefficients of the continued fraction and the differences between the convergents and $\alpha$. Using Hata's work, we show that the terms in the above formulas are the coefficients of the Legendre transform of $g$ in a certain Schauder basis, allowing us to interpret our formulas as Parseval-type identities. We hope that the Legendre duality sheds new light on Hata's approach. Raising the terms in the above summation formula to the power $s$ we obtain a function $F_g(s)$. We prove that for a strictly concave $g$, the function $F_g(s)$ converges for $s>2/3$ and diverges at $s=2/3$.