Topics in the distribution of farey sequences

Abstract

In this thesis, we study the distribution of Farey sequences. Let Q be a positive integer. The Farey sequence FQ of order Q is the set of irreducible fractions in [0, 1] whose denominators do not exceed Q. The first study characterizing the behavior of sequences via equidistribution was carried out in the seminal paper of Weyl. Equidistribution refers to being evenly spaced in a measure space. Equidis- tributed sequences are particularly useful for performing numerical integration. The notion of equidistribution does not provide information about finer statistics, such as randomness, local clustering, and periodic structure of sequences. To study the fine-scale statistics of a sequence, one can study the nearest neighbor gap distri- bution, as well as ν-level correlation measure. We study the equidistribution and correlation measure for Farey sequences. The study of the Farey sequence is of independent interest because of its role in the Diophantine approximation, the circle method, and its connection to the Rie- mann Hypothesis as established by the classical work of Franel and Landau. The Farey fractions of order Q have a one-to-one correspondence with visible lattice points in the triangle with vertices (0, 0), (0, Q), and (Q, Q) through straight lines passing through the origin. The visible lattice points along polynomials have been introduced and studied by Chaubey et al. Motivated by this, we introduce polyno- mial Farey fractions as a subset of fractions a/q ∈ [0, 1] such that the point (a, q) is visible through polynomial curves and examine their distribution. In particular, we study and prove that the lim sup of the pair correlation measure of the polynomial Farey sequence is bounded. For the specific polynomial P (x) = x(x + 1), we show. that the pair correlation measure exists and establish an explicit formula for the pair correlation function which is non-Poissonian. Further, when restricting to prime denominators, the pair correlation measure is shown to be Poissonian. A sequence is said to be Poissonian if it behaves like a random uniformly distributed sequence. It is interesting to study the distribution of Farey fractions with denominators in arithmetic progression, as it is closely related to the Generalized Riemann Hypothesis. Moreover, we study an analog of Chebyshev’s bias question for polynomial Farey fractions with denominators in an arithmetic progression. Chebyshev’s bias question deals with the prime number races and states that there are more primes of the form 4n + 3 than the primes of the form 4n + 1. Furthermore, we study the distribution of the sequence of Farey fractions with k-free denominators lying in an arithmetic progression, denoted by F (m) Q,k . We prove that the sequence F (m) Q,k Q is equidistributed by establishing an estimate for a Weyl sum. Additionally, we establish an equivalent criterion for the Generalized Riemann Hypothesis in terms of the distribution of fractions in F (m) Q,k analogous to the classical results of Franel and Landau. We also investigate the correlation measure of the sequence F (m) Q,k Q and provide an explicit form for the pair correlation measure. Another effective approach to understanding the distribution of the Farey fractions is to examine their indices. We study the distribution of Farey indices by deriving asymptotic formulas for the moments of the index function of Farey fractions with B-free denominators which lie in a given arithmetic progression.