Formulas and algorithms for the length of a Farey sequence

This paper proves several novel formulas for the length of a Farey sequence of order n. The formulas use different trade-offs between iteration and recurrence and they range from simple to more complex. The paper also describes several iterative algorithms for computing the length of a Farey sequence based on these formulas. The algorithms are presented from the slowest to the fastest in order to explain the improvements in computational techniques from one version to another. The last algorithm in this progression runs in $$O(n^{2/3})$$ O ( n 2 / 3 ) time and uses only $$O(\sqrt{n})$$ O ( n ) memory, which makes it the most efficient algorithm for computing $$|F_n|$$ | F n | described to date. With this algorithm we were able to compute the length of the Farey sequence of order $$10^{18}$$ 10 18 .

Farey Sequences for Thin Groups

The Farey sequence is the set of rational numbers with bounded denominator. We introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete and 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 and the gap distribution converges and answer an associated problem in Diophantine approximation. Moreover, for one example, we derive an explicit formula for the gap distribution. For this example, we construct the analogue of the Gauss measure, which is ergodic for the Gauss map. This allows us to prove a theorem about the associated Gauss–Kuzmin statistics.

Sign of the Mertens function

Lehman proved that the sum of certain Mertens function values is 1. Functions involving the sum of the signs of these Mertens function values are considered here. Specifically, the upper bounds of these functions involving the number of Mertens function values equal to zero are determined. 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 relationship between the Mertens function and the Farey sequence. Functions of subsets of the fractions in Farey sequences that are analogous to the Mertens function are introduced. Results analogous to Lehman’s theorem are the defining property of these functions. A relationship between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem is postulated.

Extraction of Farey Sequence from Stern-Brocot Sequence

In this paper we define and discuss the properties of Farey Sequence and Stern - Brocot Tree and the principal aim of this paper is to extract Farey sequence from Stern - Brocot Tree .

Equidistribution of Farey sequences on horospheres in covers of and applications

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

Ducci iterates and similar ordering of visible points in convex regions

Hardy et al. first introduced the notion of similar ordering of pairs of rationals, and Mayer proved that pairs of Farey fractions in [Formula: see text] are similarly ordered when [Formula: see text] is large enough. We generalize Mayer’s result to Ducci iterates of Farey sequence and visible points in certain regions in the plane. We also study the distribution of values of generalized indices of these sequences.

On Some Special Property of The Farey Sequence

In this paper, some special property of the Farey sequence is discussed. We prove in each term of the Farey sequence, the sum of elements in the denominator is two times of the sum of elements in the numerator. We also prove that the Farey sequence contains a palindrome structure.

Farey Sequence for Error Correcting Codes and Medical Images

A new scheme for Error Correcting Codes and the coding of medical images through Farey Sequence is brought in the paper. It brings out the reduced real point representations and more of integer point representations. The test case and outcomes are explained in the section 4 and 5.