the relation between Von Mangoldt function and Riemann zeta function

Question

I've seen that $\ln\left(\zeta\right)$ can be expressed in the following manner: $$\ln\left(\zeta(s)\right)=\sum_{n=2}^{∞}\frac{Λ\left(n\right)}{\ln\left(n\right)}\cdot\frac{1}{n^{s}}$$ where $Λ\left(n\right)$ is Von mangoldt function.

I know some properties of the function but how the formula can be derived? any hint or full proof is appreciated.

Answer 1

Let $S$ denote the set of prime powers, and write an arbitrary element $n$ of $S$ as $p^k$ with $p\in\Bbb P,\,k\in\Bbb N$. The left-hand side is$$-\sum_{p\in\Bbb P}\ln(1-p^{-s})=\sum_{k\in\Bbb N,\,p\in\Bbb P}\frac{p^{-ks}}{k}=\sum_{n\in S}\frac{n^{-s}\ln p}{\ln n}=\sum_{n\ge2}\frac{n^{-s}\Lambda(n)}{\ln n}.$$