# Prime number sums and Dirichlet characters

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 $$C(\chi) = \sum_{n=2}^\infty \rho(n,\chi) (\frac{1}{\log n}-\frac{1}{\log (n+1)})\qquad (= \log L(1,\chi))$$
• Can you say what is happening in the equality $O(1/log N) + \sum_{n=2}^N O(\frac{1}{n\log^2 n})$? If we skip this equality and go to the next, I see that the series defining $C(\chi)$ converges, just because $\rho$ is bounded and using this bound we have a telescoping series for the log's and hence get an upper bound of $1/\log 2$. From this I get that the series $\sum_{n=2}^\infty \frac{\chi(n) \Lambda(n)}{n \log n}$ converges. How am I doing? – Jordan Nov 19 '17 at 1:33
• @Jordan Sure you can use that the sum is telescoping, but I wanted to use $\frac{1}{\log n}-\frac{1}{\log (n+1)} = \int_n^{n+1} \frac{dx}{x \log^2 x}= \mathcal{O}(\frac{1}{n \log^2 n})$ showing how Abel summation formula and summation by parts are the same thing, and which works even when $\rho$ is unbouned – reuns Nov 19 '17 at 1:46
• Also the $\mathcal{O}$ constants depend on the one for $\rho(N,\chi)= \mathcal{O}(1)$, thus I should have written $\mathcal{O}_\chi$ everywhere – reuns Nov 19 '17 at 1:48