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 $p$.Comment: 32 pages, 10 figure

Products of binomial coefficients and unreduced Farey fractions

“…Another relation of binomial products G n to the distribution of prime numbers arises via their connection to products of Farey fractions. This connection via Möbius inversion creates other products of binomial coefficients which may be directly related to the Riemann hypothesis, for which see [36]. Moreover, there are general relations known between families of integer factorial ratios and the Riemann hypothesis.…”

“…See [36,Section 3] for a further discussion of Mikolas's results, which include unconditional error bounds for R(n). The true subtlety in the Mikoläs formula seems to resolve around oscillations in the function Φ(x) of magnitude at least Ω(x √ log log x) which themselves are related to zeta zeros.…”

“…(2) For number-theoretic applications (as in [36]) one extends the values G n to positive real x another way, making it a step function setting G x := G x . For the step function definition the function log(G x ), viewed as a function of a real variable x, has jumps of size n at integer values of x.…”

“…Our results yield a Chebyshev-type estimate for π(x) and suggest the possibility of a approach to the prime number theorem via radix expansion properties of n to prime bases. In another direction, a connection of the G n to the Riemann hypothesis may exist via their relation to products of Farey fractions, see [36].…”

Partial Factorizations of Generalized Binomial Products

“…statistics A(n, x) and B(n, x). The work [21] studied analogous statistics for Farey fractions. The paper [13] of the first two authors studied the statistics A(n, x) and B(n, x) for products of binomial coefficients.…”