The fourth moment of the range and Parkinson's efficiency

Starting from Feller's density $f_R(r)=8\sum_{k\ge1}(-1)^{k-1}k^2\varphi(kr)$ for the range of standard Brownian motion on $[0,1]$: (a) Derive $E[R^4]=9\zeta(3)$, justifying the interchange of sum and integral. (b) Deduce the variance of the per-day Parkinson estimator and its efficiency relative to $r_t^2$. (c) How many days of Parkinson estimates give the same standard error of $\hat\sigma^2$ as 252 days of squared close-to-close returns? What is the relative standard error of $\hat\sigma$ from 20 days of each?