Products of binomial coefficients and unreduced Farey fractions

Abstract

This paper studies the product $\bar{G}_n$ of the binomial coefficients in the n-th row of Pascal's triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order n. It studies its size as a real number, measured by its logarithm $log(\bar{G}_n)$, and its prime factorization, measured by the order of divisibility by a fixed prime p, each viewed as a function of n. It derives three formulas for its prime power divisibility, $ord_p(\bar{G}_n)$, two of which relate it to base p radix expansions of n, and which display different facets of its behavior. These formulas are used to determine the maximal growth rate of each $ord_p(\bar{G}_n)$ and structure of the fluctuations of these functions. It also defines analogous functions for all integer bases $b$ replacing prime bases. A final topic relates the factorizations of $\bar{G}_n$ to Chebyshev-type prime-counting estimates and the prime number theorem.