Dirichlet's Theorem on Arithmetic Sequences/Lemma 2

Lemma for Dirichlet's Theorem on Arithmetic Progressions
Let $a, q$ be coprime integers.

Let $\mathcal P_{a, q}$ be the set of primes $p$ such that $p \equiv a \pmod q$.

Define:


 * $\eta_{a, q} : n \mapsto \begin{cases}

1 & : n \equiv a \pmod q\\ 0 & : \text{otherwise} \end{cases}$

Let $G = \left({\Z / q \Z}\right)^\times$.

Let $G^*$ be the dual group of characters on $G$.

Then for all $n \in \N$:


 * $\displaystyle \eta_{a, q} \left({n}\right) = \sum_{\chi \mathop \in G^*} \frac {\overline \chi \left({a}\right)} {\phi \left({q}\right)} \chi \left({n}\right)$

Proof
There is only one $x \in G$ such that $\eta \left({x}\right) \ne 0$, and this equals $\eta \left({a}\right) = 1$.

So:


 * $\displaystyle \sum_{x \mathop \in G} \eta_{a, q} \left({x}\right) \overline \chi \left({x}\right) = \overline \chi \left({a}\right)$

Therefore, by Discrete Fourier Transform on Abelian Group we have for all $x \in G$:


 * $\displaystyle \eta \left({x}\right) = \frac 1 {\phi \left({q}\right)} \sum_{\chi \mathop \in G^*} \overline \chi \left({a}\right) \chi \left({x}\right)$

as required.