On this website, within a proof of the prime number theorem, they boldly make the following claim:$$-\frac{\zeta'(x)}{\zeta(x)}=\sum_n \frac{\Lambda(n)}{n^x}$$
Where $\Lambda(x)$ is the von Mangoldt function.
Without anything to back it up. From some research, I can see that this identity is quite common and is related to Dirichlet Series, but I am yet to find a proof.
How can we prove this identity?
For any $s>1$ we have $$ \zeta(s)=\prod_{p\in\mathcal{P}}\left(1-\frac{1}{p^s}\right)^{-1} \tag{1}$$ then $$ \frac{\zeta'(s)}{\zeta(s)}=\frac{d}{ds}\log\zeta(s) = -\sum_{p\in\mathcal{P}}\frac{\log(p)}{p^{s}-1}=-\sum_{n\geq 1}\frac{\Lambda(n)}{n^s}\tag{2} $$ by just considering that $\frac{1}{p^s-1} = \frac{1}{p^{s}}+\frac{1}{p^{2s}}+\frac{1}{p^{3s}}+\ldots$ and rearranging.