I have the following sum that I am trying to put into a cleaner formula (one that I can hopefully find the value of). This looks like it may be similar to a Riemann Sum, so it could turn into an integral. Any ideas on how to evaluate it? $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k = 1}^{n}\left(\frac{1}{2}-\frac{1}{4}\frac{\log \log p_k }{\log p_k} \right)$$
$p_k$ is the $k^{th}$ prime.