Products of Farey Fractions

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(F_n), 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.

KEYWORDS:

• Farey sequence
• unreduced Farey sequence

2000 AMS Subject Classification:

• Primary 11K55
• Secondary 11S82

Acknowledgments

We thank J. Arias de Reyna, R. C. Vaughan, and the two reviewers for helpful comments on this article. The first author thanks Harm Derksen for bringing up questions on Farey products, resulting in [Derksen and Lagarias 11a, Derksen and Lagarias 11b]. Work of H. Mehta on this project started as part of an REU program at the University of Michigan, with the first author as mentor.^{Figure A.1}

Funding

Work of the first author was supported by NSF Grants DMS-1101373 and DMS-1401224.

Notes

¹Here f(x) = Ω_±(g(x)) means there is a positive constant such that infinitely often f(x) > c|g(x)| and infinitely often f(x) < −c|g(x)|.

²The data in Figure 5 seem insufficient to discriminate between growth of order N and of order Nlog_pN. For N = p² − 1, the quantity log_pN ≈ 2 is approximately constant.

³Here Φ_∞(n) is the “replacement main term” mentioned in Sect. 1.2 and defined in (5–2(5--2) ).