Considering a non-$\chi_0$ Dirichlet character $\chi:\mathbb{N}\to\mathbb{U}_n$ for a given fixed $n\in\mathbb{N}^\star$, and using the relation $\displaystyle\sum_{k=0}^{n-1}\chi(k)=0$ (which is obtained by summing all over $\mathbb{Z}/n\mathbb{Z}$), one can use the Dirichlet's test to prove the convergence of the series $\displaystyle\sum_{k\geqslant1}\frac{\chi(k)}k$. My goal is to show that the sum of the series is non-zero.
I tried several things. At first, mimicing the Abel's transformation to the series is not working. Then, I thought about writing $\chi(k)=e^{2i\pi\xi(k)/n}$ where $\xi:\mathbb{N}\to[\![0,n-1]\!]$ is $n$-periodic, to consider, for a given upper bound $K\in\mathbb{N}^\star$ : $$\Im\left(\sum_{k=1}^K\frac{\chi(k)}k\right)=\sum_{k=1}^K\frac{\sin(2\pi\xi(k)/n)}k$$ Then, my idea was to consider an even bound $K=2\tilde{K}$, so I could hope to re-write the series up to $2K$ as a series of a difference of two positive ters up to the bound $\tilde{K}$ (I was hoping that there were as many positive and negative sine values).
I'm not necessarily asking for a complete solution, some clues would help me a lot ! Thanks already for your replies !
Kindly