Asymptotic expansions of weighted prime power counting functions

We prove several asymptotic continued fraction expansions of $\pi(x)$, $\Pi(x)$, $\operatorname{li}(x)$, $\operatorname{Ri}(x)$, and related functions, where $\pi(x)$ is the prime counting function, $\Pi(x) = \sum_{k = 1}^\infty \frac{1}{k}\pi(\sqrt[k]{x})$ is the Riemann prime counting function, and $\operatorname{Ri}(x) = \sum_{k=1}^\infty \frac{ \mu(k)}{k} \operatorname{li}(\sqrt[k]{x})$ is Riemann's approximation to the prime counting function. We also determine asymptotic continued fraction expansions of the function $\sum_{p \leq x} p^s$ for all $s \in \mathbb{C}$ with $\operatorname{Re}(s) > -1$, and of the functions $\sum_{a^x < p \leq a^{x+1}} \frac{1}{p}$ and $\log \prod_{a^x < p \leq a^{x+1}} (1 -1/p)^{-1}$ for all real numbers $a > 1$. We also determine the first few terms of an asymptotic continued fraction expansion of the function $\pi(ax)-\pi(bx)$ for $a > b > 0$. As a corollary of these results, we determine the best rational approximations of the "linearized" verions of these various functions.