Let $\chi$ be a nontrivial Dirichlet character modulo $m$. I am reading the argument in Davenport where he proves that $\sum_p \frac{\chi(p)}{p}$ converges.
I can establish after some work that $\sum_{p \leq x} \frac{\chi(p) \log p}{p} = O(1)$. I'm looking for someone to spell out if I can get from this to proving that $\sum_p \frac{\chi(p)}{p}$ converges.
Sure. Let $\Lambda(n) = \log p$ if $n=p^k$, $\Lambda(n) = 0$ otherwise, and $$\rho(N,\chi) = \sum_{n \le N} \frac{\chi(n)\Lambda(n)}{n} $$
From $\rho(N,\chi)= \mathcal{O}(1)$ for $\chi$ non-principal (a consequence of $L(1+it,\chi) \ne 0$)
and since $$\frac{1}{\log n}-\frac{1}{\log (n+1)} = \int_n^{n+1} \frac{dx}{x \log^2 x}= \mathcal{O}(\frac{1}{n \log^2 n})$$
using partial summation we obtain $$\sum_{n=2}^N \frac{\chi(n) \Lambda(n)}{n\log n} = \frac{\rho(N,\chi)}{\log N} + \sum_{n=2}^{N-1} \rho(n,\chi) (\frac{1}{\log n}-\frac{1}{\log (n+1)}) \\= \mathcal{O}(\frac{1}{\log N}) + \sum_{n=2}^{N-1} \mathcal{O}(\frac{1}{n \log^2 n}) = C(\chi)+\mathcal{O}(\frac{1}{ \log N})$$ Where $$C(\chi) = \sum_{n=2}^\infty \rho(n,\chi) (\frac{1}{\log n}-\frac{1}{\log (n+1)})\qquad (= \log L(1,\chi))$$