Stuck on this character exercise (6.13) from Apostol's Intro to Analytic Number Theory. Can anyone lend a hand? Let $f_1, \ldots, f_m$ be the characters of a finite group $G$ with order $m$, and let $a$ be an element of $G$ with order $n$. Theorem 6.7 shows us that each number $f_r(a)$ is an $n$th root of unity. Prove that every $n$th root of unity occurs equally often among the numbers $f_1(a), \ldots, f_m(a)$. 
They also give the hint:
Evaluate the sum 
$$\sum_{r=1}^m \sum_{k=1}^n f_r(a^k)e^{-\frac{2\pi i k}{n}}$$
in two ways to determine the number of times $e^{\frac{2\pi i}{n}}$ occurs.
I think evaluating in the order that it is currently written gives me 0, but I'm not sure how that helps me answer the problem. 
Here is a link to an image of the problem
 A: We need the orthogonality relations for characters, specifically
$$\sum_{r = 1}^m f_r(b) = \begin{cases} m &\text{if } b = 1 \\ 0 &\text{if } b \neq 1 \end{cases}$$
and the formula for a geometric sum, specifically
$$\sum_{k = 1}^n \rho^k = \begin{cases} n &\text{if } \rho = 1 \\ 0 &\text{if } \rho \neq 1 \end{cases}$$
for an $n^{\text{th}}$ root of unity $\rho$.
Thus, since $f_r(a^k) = f_r(a)^k$ and $f_r(a)e^{-\frac{2\pi i}{n}}$ is an $n^{\text{th}}$ root of unity, for each $r$ we have
$$\sum_{k = 1}^n f_r(a^k)e^{-\frac{2\pi ik}{n}} = \begin{cases} n &\text{if } f_r(a) = e^{\frac{2\pi i}{n}} \\ 0 &\text{otherwise}\end{cases}$$
and consequently
$$\sum_{r = 1}^m \sum_{k = 1}^n f_r(a^k)e^{-\frac{2\pi ik}{n}} = n\cdot \#\bigl\{r : f_r(a) = e^{\frac{2\pi i}{n}}\bigr\}.$$
Changing the order of summation, we have
$$\sum_{r = 1}^m f_r(a^k) = \begin{cases} m &\text{if } k = n\\ 0 &\text{if } k < n \end{cases}$$
and therefore
$$\sum_{k = 1}^n \sum_{r = 1}^m f_r(a^k)e^{-\frac{2\pi ik}{n}} = \sum_{r = 1}^m f_r(a^n)e^{-\frac{2\pi in}{n}} = m.$$
It follows that
$$\#\bigl\{r : f_r(a) = e^{\frac{2\pi i}{n}}\bigr\} = \frac{m}{n}.$$
No special properties of $e^{\frac{2\pi i}{n}}$ were used, only that it is an $n^{\text{th}}$ root of unity, thus the same argument works for
$$\sum_{r = 1}^m \sum_{k = 1}^n f_r(a^k)\rho^{-k}$$
where $\rho$ is any $n^{\text{th}}$ root of unity.
