Asymptotic Distribution of the Distance Function to the Farey Points

✍ Scribed by Pavel Kargaev; Anatoly Zhigljavsky

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 337 KB
• Volume: 65
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2110

✦ Synopsis

Let x be a real number in [0, 1], F n be the Farey sequence of order n and \ n (x) be the distance between x and F n . Assuming that n Ä we derive the asymptotic distributions of the functions n 2 \ n (x) and n\ n (x$Ân), 0 x$ n. We also establish the asymptotics for 1 0 \ $ n (x) dx, for all real $.

1997 Academic Press

1. INTRODUCTION: STATEMENT OF THE PROBLEM AND FORMULATION OF THE MAIN RESULTS

Let x be a real number in [0, 1] and F n be the Farey sequence of order n, that is, the collection of all rationals pÂq with p q, ( p, q)=1 and the denominators q n. In the present work we derive two asymptotic distributions for

the distance function between x and F n , and establish the asymptotics for 1 0 \ $ n (x) dx, for all real $. It is well-known that the elements of the Farey sequence F n are uniformly distributed asymptotically, when n Ä , and this has important consequences in number theory: for example, the Riemann hypothesis can be formulated in terms of the rate of convergence of F n to the uniform distribution, see [1,2,3]. However, little is known about other asymptotic properties of F n and the distance function \ n (x).