Ducci iterates and similar ordering of visible points in convex regions

Hardy et al. first introduced the notion of similar ordering of pairs of rationals, and Mayer proved that pairs of Farey fractions in [Formula: see text] are similarly ordered when [Formula: see text] is large enough. We generalize Mayer’s result to Ducci iterates of Farey sequence and visible points in certain regions in the plane. We also study the distribution of values of generalized indices of these sequences.

Products of binomial coefficients and unreduced Farey fractions

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.