F. P. Boca, C. Cobeli, A. Zaharescu
We prove some asymptotic results on the distribution of h-tuples of consecutive Farey fractions of order Q with odd denominators as $Q\to \infty$.
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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.
Pair correlation of Farey fractions with square-free denominators
本文研究分母为无平方因子数的 Farey 分数的对关联问题，证明其极限对关联函数存在并给出显式公式，推进了数论中 Farey 分数分布性质的相关研究。
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].
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,…
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.
Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers
该研究针对带系数互素限制的弗罗贝尼乌斯开放问题，证明此类问题仍存在无法表示的最大整数，并探究其与斯特恩序列等的关联及相关性质。
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.
On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions
研究连续SL(2,N)-饱和 Farey 分数的分母在特定区域的稠密性，并给出其分布公式。
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…
We discuss some special property of the Farey sequence. We show in each term of the Farey sequence, ratio of the sum of elements in the denominator and the sum of elements in the numerator is exactly two. We also show that the Farey sequence contains a palindromic structure.
Distribution of Farey fractions with $k$-free denominators
该研究探讨带k自由分母的Farey分数序列的分布与局部统计性质，证明其模1均匀分布，推导广义黎曼假设的等价形式，建立高阶相关测度公式，为相关数论问题提供新结果。
On the distribution of polynomial Farey points and Chebyshev's bias phenomenon
该文研究多项式Farey分数的分布与Chebyshev偏差问题，通过偏差和配对关联测度刻画其全局与局部分布，证明均匀分布性，给出特定多项式的配对关联函数显式公式，并得到计数函数误差项的Ω结果。
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…
Smallest denominators
该论文将Chen-Haynes定理及Kruwswijk-Meijer猜想推广至高维，用多维Farey空隙统计刻画最小分母分布，证明分布函数与有限矩收敛，并推广Kargaev-Zhigljavsky矩结果，获得鸽笼统计新结果。
The classical Farey sequence of height $Q$ is the set of rational numbers in reduced form with denominator less than $Q$. In this paper we introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete (and in particular thin) subgroups, they can be used to study interesting number theoretic sequences - for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute, that the gap distribution converges, and we answer an associated problem in Diophantine appr…
The number of solutions of the diophantine equation $\sum_{i=1}^k \frac{1}{x_i}=1,$ in particular when the $x_i$ are distinct odd positive integers is investigated. The number of solutions $S(k)$ in this case is, for odd $k$: \[\exp \left( \exp \left( c_1\, \frac{k}{\log k}\right)\right) \leq S(k) \leq \exp \left( \exp \left(c_2\, k \right)\right) \] with some positive constants $c_1$ and $c_2$. This improves upon an earlier lower bound of $S(k) \geq \exp \left( (1+o(1))\frac{\log 2}{2} k^2\right)$.
Rational numbers with odd greedy expansion of fixed length
研究奇贪婪展开的有限性问题，找到了展开长度为2的既约分数、给定前缀长度为m的有理数及无穷族有理数。
A distribution related to Farey sequences -- I
本文采用Boca等人引入的方法研究Farey序列的算术性质，针对特定分母同余的分数进行染色并分析间隙分布，最终推导出D=2,3且c=0时极限比例ν(r;D,c)的显式公式。
A distribution related to Farey sequences -- II
本文沿用 Boca、Cobeli 和 Zaharescu 的方法研究 Farey 序列间隔分布，针对模 3 分母染色情形给出极限比例 ν(r;3,c) 的显式公式。
Odd-odd continued fraction algorithm
本文提出奇-奇连分数算法，其主收敛子为奇分母奇分子有理数，并证明其与最佳逼近有理数等价。
We study analogues of Minkowski's question mark function $?(x)$ related to continued fractions with even or odd partial quotients. We prove that these functions are Hölder continuous with precise exponents, and that they linearize the appropriate versions of the Gauss and Farey maps.
In this note, for each odd prime $p$, we show that the orders of the $KU$-local homotopy groups of the mod $p$ Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of $KU$-local stable homotopy groups.
Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to $h$, then the nonzero terms of the generating function are unimodal, with the integer $h$ having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerator…
We prove results concerning the joint limiting distribution of the renewal time of denominators and consecutive digits of random irrational numbers in the case of continued fractions with even partial quotients, with odd partial quotients, and for Nakada's α-expansions.
Odd denominators in the Lonely Runner spectrum for six speeds
本文针对六个速度的孤独跑者问题，证明ML<1/6的元组分母为奇数，仅出现k=1或3，还验证了相关猜想并给出精确计算代码。
The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we eventually get to 1. In a previous paper of the author the set of odd positive integers x such that R^k(x) = 1 has been characterized as the set of odd integers whose binary representation belongs to a set of strings G_k. Each string in G_k is the concatenation of k strings z_k z_{k-1} ... z_1 where each z_i is a finite and contiguous extract from some power of a st…
Odd moments and adding fractions
论文研究了短区间内互质余数分布的奇 moment 上界，并验证了Montgomery 和 Vaughan 的猜想，同时证明了 Hardy-Littlewood 猜想中奇数 k-tuples 素数数量的平均值上界。
On a tree of rational functions related to continued fractions
本文提出了一棵基于连分数的有理函数树，研究其结构特性及在正有理数范围内的唯一性。
We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over $\mathbb Q$ spanned by $1,ζ(m),ζ(m+2),...,ζ(m+2h)$, where $m$ and $h$ are integers such that $m\ge2$ and $h\ge0$. In particular, we immediately get the following results as corollaries: at least one of the eight numbers $ζ(5),ζ(7),...,ζ(19)$ is irrational, and there exists an odd integer $j$ between 5 and 165 such that 1, $ζ(3)$ and $ζ(j)$ are …
On the calculation of an integral introduced by Hubert Delange
本文研究Hubert Delange引入的积分，先给出q>1时积分的一般性质并简化问题，再针对q=2的情况提供公式，将问题转化为特定和的计算。
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