15. Let $f(n)$ be a number-theoretic function with period $k$, and denote $e(\theta)=\mathrm{e}^{2 \pi i \theta}$. Prove:
$$f(n)=\sum_{l=1}^{k} a_{l} e\left(-n \frac{l}{k}\right),$$

where $a_{l}=\frac{1}{k} \sum_{m=1}^{k} f(m) e\left(l \frac{m}{k}\right)$. In particular, when $f(n)=\chi(n ; k)$, $a_{l}=(1 / k) G(l ; \chi)$, where $G(l ; \chi)$ is given by equation (67).