Are Primitive Dirichlet Characters linearly independent.

Question

For a positive integer $N$, let $$S_N=\{ \chi~\mid~ \chi \text{ is primitive Dirichlet characters modulo }F,\text{ where } F\mid N \}.$$

I want to check the Linear independence on $S_N$. More preciesly: Let $s\in \mathbb{N}$ with $\gcd(s,N)=1$. If

$$ \sum_{\chi\in S_N} c_\chi \chi(s)=0\qquad (c_\chi\in \mathbb{C}), $$ then prove (or disprove) that the scalars $c_\chi$ are all zero.

Thanks

Answer 1

Let $G=(\mathbb Z/N\mathbb Z)^\times$ and $R=L^2(G, \mathbb C)$ the Hilbert space of functions $G \to \mathbb C$, with inner product $\left<g, h\right> = \frac{1}{|G|}\sum_{g \in G} f(g) \overline{h(g)}$. We can view $\widehat G = \hom(G, \mathbb C^\times)$ as a subset of $R$. It is elementary to check that the elements of $\widehat{G}$ are pairwise orthogonal under this inner product: this boils down to the fact that $\frac{1}{|G|}\sum_{g \in G} \chi(g) = 0$ if $\chi$ is nontrivial and $=1$ if $\chi$ is trivial. It follows that $\widehat{G}$ a linearly independent subset of $R$.