Partial Factorizations of Products of Binomial Coefficients

TL;DR

This paper investigates the asymptotic behavior of partial factorizations of the product of binomial coefficients in Pascal's triangle, revealing a quadratic growth pattern in the logarithm of these factorizations as the row index increases.

Contribution

It introduces a new analysis of the prime factorization structure of binomial coefficient products using functions based on radix expansions, with asymptotic results derived under prime number theorem assumptions.

Findings

Logarithm of partial factorizations grows as a quadratic function of n.

Asymptotic formulas are obtained for prime factor distributions in binomial products.

Results depend on prime number theorem and Riemann hypothesis assumptions.

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.