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, $or{d}_p({\bar{G}}_n)$, 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 $or{d}_p({\bar{G}}_n)$ 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.

30 pages, 3 figures, two Appendices. ; v2 is 31 pages, Appendices moved before reference list; v3 is 31 pages,corrections to match journal version

References in corpus (1)

• Euler's constant: Euler's work and modern developments

Cited by in corpus (4)

• Partial Factorizations of Products of Binomial Coefficients
• The factorial function and generalizations, extended
• Summing the sum of digits
• Spectral Polyhedra