Partial Factorizations of Products of Binomial Coefficients

Abstract: Let $G_n= \prod_{k=0}^n \binom{n}{k},$ the product of the elements of the $n$-th row of Pascal's triangle. This paper studies the partial factorizations of $G_n$ given by the product $G(n,x)$ of all prime factors $p$ of $G_n$ having $p \le x$, counted with multiplicity. It shows $\log G(n, \alpha n) \sim f_G(\alpha)n^2$ as $n \to \infty$ for a limit function $f_{G}(\alpha)$ defined for $0 \le \alpha \le 1$. The main results are deduced from study of functions $A(n, x), B(n,x),$ that encode statistics of the base $p$ radix expansions of the integer $n$ (and smaller integers), where the base $p$ ranges over primes $p \le x$. Asymptotics of $A(n,x)$ and $B(n,x)$ are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.