Generalized Logarithmic Integral

Question

Euler's logarithmic integral (of particular application in the Prime Number Theorem, for instance) is of the form \begin{equation*} \text{li}(x) := \int_0^{x} \frac{dt}{\log t} \end{equation*} and it is well-known that $\text{li}(x) \sim \frac{x}{\log x}$. There are also useful approximations like the Ramanujan series for $x > 1$: \begin{equation*} \text{li}(x) = \gamma + \log_2 x + \sum_{k\geq 1} \frac{(\log x)^k}{k\cdot k!} \end{equation*} Let $\alpha > 0$. Are there analogues of the above results for \begin{equation*} \text{li}_{\alpha}(x) := \int_2^{x} \frac{dt}{(\log t)^{\alpha}}? \end{equation*}