A Note on Farey Fractions With Odd Denominators

On the distribution of index of Farey Sequences

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.

关于连续SL(2,N)-饱和 Farey 分数分母的分母

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。

A note on the pair correlation of Farey fractions

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.

A density theorem on even Farey fractions

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…

带k自由分母的Farey分数的分布

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。

关于多项式Farey点的分布与Chebyshev偏差现象

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。

Products of Farey Fractions

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 weighted Farey sequence and a sliding section for the horocycle flow

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…

最小分母

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。

Egyptian Fractions with odd denominators

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 $S(k) \geq \exp \left( (1+o(1))\frac{\log 2}{2} k^2\right)$.

具有固定长度奇贪婪展开的有理数

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。

与Farey序列相关的一个分布——I

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。

一个与 Farey 序列相关的分布 —— II

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。

奇-奇连分数算法

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。

On Minkowski type question mark functions associated with even or odd continued fractions

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.

Denominators of special values of zeta-functions count KU-local homotopy groups of mod p Moore spectra

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.

Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to $ω(t)$

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…

On certain statistical properties of continued fractions with even and with odd partial quotients

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.

六个速度下孤独跑者谱中的奇分母

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。

奇 moment 与分数加法

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。