The relation of $\zeta$-function and $p^k$ for $Re(s) \le 1$? The von Mangoldt function:
$$\Lambda(n) = \begin{cases} \log p &; \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 &; \mbox{otherwise.} \end{cases}$$
establishes a relation between the Riemann $\zeta$-function and prime powers $p^k$ via:
$$\log \zeta(s)=-\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^s}$$
this is however for $Re(s)>1$.
Is there a way to expand the above relation to $Re(s) \le 1$?
Or is there instead another formalism that would relate between $\zeta$ to $p^k$ in such clarity in $Re(s) \le 1$?

You can differentiate both sides and get $\zeta'(s)/\zeta(s)$ = $-\sum_{n=1}^\infty \Lambda
(n)/n^s$. Now $\zeta'(s)/(\zeta(s))^2$ is analytic for $Re(s)>1/2$ if you assume Riemann Hypothesis. So you can consider $\zeta'(s)/(\zeta(s))^2$ = $\sum_{n=1}^{\infty}f(n)/n^s$ where $f(n)=-\sum_{d|n}\mu(d)\Lambda(n/d)$.
