Products of binomial coefficients and unreduced Farey fractions

Abstract

This paper studies the product [Formula: see text] of the binomial coefficients in the [Formula: see text]th row of Pascal’s triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order [Formula: see text]. It studies its size as a real number, measured by [Formula: see text], and its prime factorization, measured by the order of divisibility [Formula: see text] by a fixed prime [Formula: see text], each viewed as a function of [Formula: see text]. It derives three formulas for [Formula: see text], two of which relate it to base [Formula: see text] radix expansions of integers up to [Formula: see text], and which display different facets of its behavior. These formulas are used to determine the maximal growth rate of each [Formula: see text] and to explain structure of the fluctuations of these functions. It also defines analogous functions [Formula: see text] for all integer bases [Formula: see text] using base [Formula: see text] radix expansions replacing base [Formula: see text]-expansions. A final topic relates factorizations of [Formula: see text] to Chebyshev-type prime-counting estimates and the prime number theorem.