Dirichlet series are an important area of research in Analytic Number Theory, and their values at $s=1$ (or in general on the edge of their abscissa of convergence) are generally of special importance. I know that it is not true in general that
\begin{equation} \sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}\tag{1} \end{equation}
since if so there would be a very simple proof of the PNT from just applying this to $a_n=\mu(n)$. My question is, under what conditions does (1) hold?
I can show that if $\sum_{n=1}^{\infty}\frac{a_n}{n}$ converges then (1) must hold using a very simple $\epsilon-\delta$ proof, but I can't find any broader statements. The ideal condition that I would like to show is that if the partial sums $\sum_{n=1}^{N}\frac{a_n}{n}$ are bounded then then (1) must hold. I don't know how I would go about proving this though, and any insights on this general area would be greatly appreaciated.