Divisibility of the central binomial coefficient $\binom {2n}{n}$

Abstract:

We show that for every fixed $\ell \in \mathbb {N}$, the set of $n$ with $n^\ell |\binom {2n}{n}$ has a positive asymptotic density $c_\ell$ and we give an asymptotic formula for $c_\ell$ as $\ell \to \infty$. We also show that $\# \{n\leqslant x, (n,\binom {2n}{n})=1 \} \sim cx/\log x$ for some constant $c$. We use results about the anatomy of integers and tools from Fourier analysis. One novelty is a method to capture the effect of large prime factors of integers in general sequences.