Related papers: A note on the pair correlation of Farey fractions

In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example,…

Recently, Andrews and Bachraoui considered a generating function $F_{k,m}(q)$ associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for $m=1$. They showed this property for $1 \leq k…

We prove some asymptotic formulae concerning the distribution of the index of Farey fractions of order Q as $Q\to \infty$.

In this paper we generalize some of our results from, `A note on Farey fractions with odd denominators' to subsets of Farey fractions consisting of fractions with denominators not divisible by a given prime. We also investigate the joint…

Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs…

We prove some asymptotic results on the distribution of h-tuples of consecutive Farey fractions of order Q with odd denominators as $Q\to \infty$.

The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the…

In this paper, we prove a conjecture by Daniele Mundici on the sum of squared distances between consecutive elements in the $Q$-th Farey sequence for $Q\in\mathbb{Z}$ and $Q\geq 2$.

We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections…

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where…

The sequence $({\mathscr S}_Q)_Q$ of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $\bar{a}$ is the…

Franel and Landau derived an arithmetic statement involving the Farey sequence that is equivalent to the Riemann hypothesis. Since there is a relationship between the Mertens function and the Riemann hypothesis, there should be a…

Given $m, n, q\in \mathbb{N}$ such that $q$ is a prime power and $m\geq 3$, $a\in \mathbb{F}_q$, we establish a sufficient condition for the existence of primitive pair $(\alpha, f(\alpha))$ in $\mathbb{F}_{q^m}$ such that $\alpha$ is…

For a set $M$ of $m$ elements, we define a decreasing chain of classes of normalized monotone-increasing valuation functions from $2^M$ to $\mathbb{R}_{\geq 0}$, parameterized by an integer $q \in [2,m]$. For a given $q$, we refer to the…

We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the…

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes…

A new matrix operation based on inserting columns and rows, similarly to the mediant operation between fractions, gives rise to the Farey determinants matrix or, equivalently, the matrix of the numerators of the differences of Farey…

Let q>1 and m>0 be relatively prime integers. We find an explicit period $\nu_m(q)$ such that for any integers n>0 and r we have $[n+\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q),…

We introduce a $q$-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the $q$-rational numbers of Morier-Genoud and…

For any rational functions with complex coefficients $A(z), B(z)$ and $C(z)$, where $A(z)$, $C(z)$ are not identically zero, we consider the sequence of rational functions $H_{m}(z)$ with generating function $\sum…