I reading a book for a Seminar, where I have to present some parts out of it. In the book the other says the formula below can be proved by partial integration, but it is not clear for me, hwo that works.
$-\frac{\zeta^{\prime}(s)}{\zeta(s)}=\frac{s}{s-1}+s\int\limits_{1}^{\infty}(\psi(t)-t)t^{-s-1}dt$
The only other formula before this in this chapter of the book is
$-\frac{\zeta^{\prime}(s)}{\zeta(s)}-\zeta(s)=\sum\limits_{n=1}^{\infty}(\Lambda(n)-1)n^{-s}$
Anyone has an idea hwo to get he first formula out of this. Or u have any other way to get the first formula?