Products of Farey Fractions

TL;DR

This paper investigates the growth and divisibility properties of products of Farey fractions, exploring their potential connection to the Riemann hypothesis through theoretical and empirical analysis.

Contribution

It introduces new arithmetic functions related to Farey fraction products and proposes possible links between their properties and the Riemann hypothesis.

Findings

Growth of log(F_n) relates to the Riemann hypothesis

Properties of ord_p(F_n) are empirically studied and formulated

Evidence suggests a connection between ord_p(F_n) and the Riemann hypothesis

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 $p$.