Products of Farey Fractions

Abstract

The {Farey fractions} $F_n$ of order $n$ consist of all fractions $\frac{h}{k}$ in lowest terms lying in the closed unit interval and having denominator at most $n$. This paper considers the products $F_n$ of all nonzero Farey fractions of order $n$. It studies their growth measured by $\log(F_n)$ and their divisibility properties by powers of a fixed prime, given by $ord_p(F_n)$, as a function of $n$. The growth of $\log(F_n)$ is related to the Riemann hypothesis. This paper theoretically and empirically studies the functions $ord_p(F_n)$ and formulates several unproved properties (P1)-(P4) they may have. It presents evidence raising the possibility that the Riemann hypothesis may also be encoded in $ord_p(F_n)$ for a single 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 M\"obius inversion. It involves new arithmetic functions which mix the M\"obius function with functions of radix expansions to a fixed prime base $p$.