Products of binomial coefficients and unreduced Farey fractions

TL;DR

This paper investigates the properties and prime factorizations of the product of binomial coefficients in Pascal's triangle, linking it to Farey fractions, base expansions, and prime number theory, revealing new formulas and growth behaviors.

Contribution

It introduces new formulas for the prime power divisibility of the product of binomial coefficients, relating them to base p expansions and prime number estimates, advancing understanding of their structure.

Findings

Derived three formulas for prime power divisibility of the product

Connected divisibility properties to base p radix expansions

Linked factorizations to prime counting estimates

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.