Partial Factorizations of Products of Binomial Coefficients

Abstract

Let $G_n={\prod }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right),$ 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.

27 pages, 4 figures Submitted to International Journal of Number Theory

References in corpus (4)

• Euler's constant: Euler's work and modern developments
• Distribution of the sum-of-digits function of random integers: a survey
• Products of binomial coefficients and unreduced Farey fractions
• Products of Farey Fractions

Cited by in corpus (1)

• The factorial function and generalizations, extended