Riemann's prime counting function is given as

$$J(n)=\sum_{k=1}^{\infty}\frac{\mu(k)}{k}\operatorname{li}(n^{1/k})$$

the approximations

\begin{align} \operatorname{li}(n)\sim J(n)\tag{1}\\ \operatorname{li}(n)-\sqrt{n}/\log n\sim J(n)\tag{2}\\ (1-\sqrt{n}/\log n)\operatorname{li}(n+\sqrt{n}\log n)\sim J(n)\tag{3}\\ \end{align}

get increasingly closer. Could these be the first few terms of an asymptotic expansion of $J(n),$ or can $(3)$ not be taken any further?