Products of Farey Fractions. Issue 1 (2nd January 2017)

• Record Type: Journal Article
• Authors: Lagarias, Jeffrey C.; Mehta, Harsh
• Is Part Of: Experimental mathematics. Volume 26:Issue 1(2017)
• Page Start: 1
• Page End: 21
• Publication Date: 2017-01-02
• Subjects: Farey sequence -- unreduced Farey sequence; Primary 11K55 -- Secondary 11S82; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2015.1020578
• Languages: English
• ISSNs: 1058-6458

Abstract

ABSTRACT: The Farey fractions of order n consist of all fractions in lowest terms lying in the closed unit interval and having denominator at most n . This article considers the products F n of all nonzero Farey fractions of order n . It studies their size, given by log(Fn ), and their divisibility properties by powers of a fixed prime, given by ord p ( F n ), as a function of n . It presents evidence suggesting that information related to the Riemann hypothesis may be encoded in functions related to ord p ( F n ) for a single fixed prime p . This encoding makes use of a relation of these products to the products G n of all reduced and unreduced Farey fractions of order n, which are connected by Möbius inversion. It introduces new arithmetic functions which mix the Möbius function with functions of radix expansions to a fixed prime base p .