I am trying to prove convergence of certain series related to non-principal Dirichlet series. In the proof, I want to use the following fact: $$ \sum_{n\leq x} \frac{\chi(n)\Lambda(n)}{n} \tag{1} $$ converges as $x\to\infty$. Here $\chi$ is some non-principal character (say, mod $k$) and $\Lambda$ is the von Mangoldt function.
The only proof of convergence of (1) I know follows from Lemmas 7.3-7.8 in Apostol's Introduction to Analytic Number Theory. The proof feels a little bit long-winded for me (maybe because I do not understand it well), so here is my question:
How would one directly prove that (1) converges ?