What would be the rate of convergence of:
$$\sum^N_n \frac{\mu(n) \log(n)}{n}$$
?
I know that as $N \to \infty$ the series approaches to $-1$, but I am not able to get how fast it does converge using big-O notation. Any help?
Edit 1: as noted in the comments, this series equals
$$\lim_{s\to 1^+} \frac{\zeta'(s)}{\zeta^2(s)}$$
Thank you.