Let $$Li(x)=\int_2^x\frac{dt}{\log t},$$ the logarithmic integral and we define for $\Re s>1$ the complex function $$f(s)=s\int_2^\infty\frac{Li(u)}{u(u^s-1)}du.$$
Question. I say that previous expression defines $f(s)$ as an analytic function for $\Re s>1$. Can you discuss if the analytic continuation of $f(s)$ has pole/s, and what is its residue? (Please if my question isn't well possed explain why.)
There is in the literature the corresponding identity defining $\log\zeta(s)$, with the prime-counting function $\pi(u)$ in the integrand.
Thanks in advance.