Logarithmic Derivative of Riemann Zeta Function

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


 * $\displaystyle\zeta(s) = \sum_{n \geq 1} n^{-s},\qquad (\Re(s) > 1)$

Then for all $s$ with $\Re(s) > 1$ we have


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

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

Proof
By Equivalence of Definitions of Riemann Zeta Function, we have:


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

where $p$ ranges over the primes. Therefore, using Laws of Logarithms


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

Taking the derivative, we obtain

Also, we have by the definition of $\Lambda$,


 * $\displaystyle \sum_{n \geq 1} \Lambda(n)n^{-s} = \sum_{p} (\log p) \sum_{n \geq 1} p^{-ns}$

so we are done.