Logarithmic Derivative of Riemann Zeta Function

Theorem
Let $\zeta$ be the Riemann zeta function:


 * $\displaystyle \forall s \in \C: \Re \left({s}\right) > 1: \zeta \left({s}\right) = \sum_{n \mathop \ge 1} n^{-s}$

Then for all $s$ with $\Re \left({s}\right) > 1$:


 * $\displaystyle -\frac{\zeta' \left({s}\right)} {\zeta \left({s}\right)} = \sum_{n \ge 1} \Lambda(n) n^{-s}$

where $\Lambda$ is von Mangoldt's function.

Proof
By definition of the Riemann zeta function:


 * $\displaystyle \zeta \left({s}\right) = \prod_p \frac 1 {1 - p^{-s}}$

where $p$ ranges over the primes.

From Laws of Logarithms:


 * $\displaystyle \ln \zeta \left({s}\right) = - \sum_p \ln \left({1 - p^{-s} }\right)$

Taking the derivative:

Also, by the definition of $\Lambda$:


 * $\displaystyle \sum_{n \mathop \ge 1} \Lambda \left({n}\right) n^{-s} = \sum_p \left({\ln p}\right) \sum_{n \mathop \ge 1} p^{-n s}$

and the proof is complete.