Partial factorizations of products of binomial coefficients

Abstract: Let $G_n$ denote the product of the binomial coefficients in the $n$-th row of Pascal's triangle. Then $\log G_n$ is asymptotic to $\frac{1}{2}n^2$ as $n \to \infty$. Let $G(n,x)$ denote the product of the maximal prime powers of all $p \le x$ dividing $G_n$. We determine asymptotics of $\log G(n, \alpha n) \sim f(\alpha)n^2$ as $n \to \infty$, with error term. Here \[ f(\alpha) = \frac{1}{2} -\alpha \left\lfloor \frac{1}{\alpha} \right\rfloor + \frac{1}{2} \alpha^2 \left\lfloor \frac{1}{\alpha}\right\rfloor^2 + \frac{1}{2} \alpha^2 \left\lfloor \frac{1}{\alpha} \right\rfloor \] for $0< \alpha \le 1$. The result is based on analysis of associated radix expansion statistics $A(n,x)$ and $B(n,x)$. The estimates relate to prime number theory, and vice versa.

Joint work with Lara Du.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.

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