Let $F_Q$ be the set of Farey fractions of order $Q$. Given the integers $\d\ge 2$ and $0\le \c \le \d-1$, let $F_Q(c,d)$ be the subset of $F_Q$ of those fractions whose denominators are $\equiv c \pmod d$, arranged in ascending order. The problem we address here is to show that as $Q\to\infty$, there exists a limit probability measuring the distribution of $s$-tuples of consecutive denominators of fractions in $F_Q(c,d)$. This shows that the clusters of points $(q_0/Q,q_1/Q,...,q_s/Q)\in[0,1]^{s+1}$, where $q_0,q_1,...,q_s$ are consecutive denominators of members of $F_Q$ produce a limit set, denoted by $D(c,d)$. The shape and the structure of this set are presented in several particular cases.
As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in arithmetic progression.
In this paper we examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of values of numerators of differences of consecutive elements in this set. After proving an asymptotic result for these frequencies, we use estimates coming from incomplete Kloosterman sums to generalize our result to subintervals of [0,1].
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 distribution of numerators of differences of h-tuples of consecutive fractions in these sets.
In this article, we study the distribution of index of Farey fractions which was first introduced and studied by Hall and Shiu. We provide asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with $\mathcal{B}$-free denominators. Additionally, we reconsider the squarefree case earlier done in [ALVZ08], and obtain new results for moments of indices with square-free denominators. We also study higher level correlations of the index function generalizing earlier known results on two level correlations.
The pair correlations of Farey fractions with denominators $q$ satisfying $(q,m)=1$, respectively $q\equiv b \pmod{m}$ with $(b,m)=1$, are shown to exist and are explicitly computed.
Let $F_Q$ be the Farey sequence of order $Q$ and let $F_{Q,o}$ and $F_{Q,e}$ be the set of those Farey fractions of order $Q$ with odd, respectively even denominators. A fundamental property of $F_Q$ says that the sum of denominators of any pair of neighbor fractions is always greater than $Q$. This property fails for $F_{Q,o}$ and for $F_{Q,e}$. The local density, as $Q\to\infty$, of the normalized pairs $(q'/Q,q''/Q)$, where $q',q''$ are denominators of consecutive fractions in $F_{Q,o}$, was computed previously. The density increases over a series of quadrilateral steps ascending in a harmo…
The {Farey fractions} $F_n$ of order $n$ consist of all fractions $\frac{h}{k}$ in lowest terms lying in the closed unit interval and having denominator at most $n$. This paper considers the products $F_n$ of all nonzero Farey fractions of order $n$. It studies their growth measured by $\log(F_n)$ and their divisibility properties by powers of a fixed prime, given by $ord_p(F_n)$, as a function of $n$. The growth of $\log(F_n)$ is related to the Riemann hypothesis. This paper theoretically and empirically studies the functions $ord_p(F_n)$ and formulates several unproved properties (P1)-(P4) t…
The Farey sequence is the sequence of all rational numbers in the real unit interval, stratified by increasing denominators. A classical result by Hall says that its normalized gap distribution is the same as the distribution of the random variable 1/(2 zeta(2) xy) on a certain unit triangle. In this paper we weight the denominators by an arbitrary piecewise-smooth continuous function, and we characterize the resulting gap distribution as that of a multiple of the above variable, defined on a certain unit pentagon. Our characterization refines previous results by Boca, Cobeli and Zaharescu, bu…
There are two questions in analytic number theory which have attracted much attention over the years. The first one is about the asymptotic formula for the variance associated with the distribution of a real sequence in arithmetic progressions, which origins in the work of Barban. The second one is about the function $r_3(n)$, the number of ordered representation of $n$ in the sum of three positive cubes, and several results are recently proved by Robert C. Vaughan. The paper is concerned with the conjunction of these questions, with the application of the Hardy-Littlewood Method and the Farey…
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.
We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first main result is a necessary and sufficient criterion (applicable to all but finitely many primes) for determining when the coefficients of a hypergeometric series with rational parameters are p-adically unbounded. This criterion is then used to show that the set of unbounded primes for a given series is, up to a finite discrepancy, a finite union of primes in arithmetic progressions. This set can be computed explicitly. We characterize when the density of th…
Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity, the proportion of the $r$-tuples of consecutive denominators with components in certain apriori fixed arithmetic progressions with the same ratio approaches a limit, which is independent on the interval. The limit is given explicitly and it is completely described in a few particular instances.
Research Center in Mathematics and Applications (CIMA)（数学与应用研究中心）；Department of Mathematics, School of Sciences and Technology of the University of Évora（埃武拉大学科学与技术学院数学系）
本文研究了素数表示常数递归构造中出现的映射$\mathcal{T}(x)=\lfloor x floor(1+\{x\})$的有理动力学，定义了有理数的阶，并证明了有限阶有理数的自然密度为1，同时给出了特定分母下的显式分数家族。
For integers $m \geq 2$, we study divergent continued fractions whose numerators and denominators in each of the $m$ arithmetic progressions modulo $m$ converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a general class of Poincar{é} type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity. We also generalize a curious $q$-continued fraction of Ramanujan's …
