Legendre duality for certain summations over the Farey pairs

TL;DR

This paper introduces new summation formulas over Farey pairs linked to primitive vectors, involving Legendre duality and applications to continued fractions, with connections to classical series and convergence properties.

Contribution

It presents novel summation formulas over Farey pairs using Legendre duality, connecting geometric, number-theoretic, and functional analysis concepts.

Findings

Formulas depend explicitly on primitive vector components

Special cases recover classical series involving π

Convergence of associated functions established for s > 2/3

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 $s=2/3$.