I am currently following this course http://www.few.vu.nl/~sdn249/modularforms16/ from the University of Utrecht, and I am trying to solve problem 5.(a) from the 6th Homework Sheet.
Let $\chi$ be a primitive Dirichlet character modulo $N$, and let $\zeta$ be a $N$-th root of unity in $\mathbf{C}$. I am asked to show that $$\sum_{j=0}^{N-1}\chi(j)\frac{x+\zeta^j}{x-\zeta^j}=\frac{2N}{\tau(\overline{\chi})(x^N-1)}\sum_{m=0}^{N-1}\overline{\chi}(m)x^m\in\mathbf{C}[[x]].$$ As a hint, it says "compute the residues".
I am guessing it's because my Complex Analysis skills are quite rusty at this point, but I really don't see how to start proving this. Any ideas?
The LHS is $$F(x)=\sum_{j=0}^{N-1}\chi(j)\frac{2\zeta^j}{x-\zeta^j}$$ so equals $g(x)/(x^n-1)$ where $g$ is a polynomial of degree $\le nN-1$. This polynomial will be characterised by $$2\chi(j)\zeta^j=\lim_{x\to0}\frac{(x-\zeta^j)f(x)}{x^N-1} =\frac{\zeta f(\zeta^j)}N.$$ Putting $\zeta^j$ into $f(x)=\sum\overline{\chi}(m)x^m$ will give you something that's basically a Gauss sum.