I am trying to find a reference for the following proposition:

Let $m$ and $n$ be coprime. Then, $$ \sum_{k=0}^{r-1} \exp\left( i \frac{2\pi}{n} k m \right) = 0 $$ if and only if $r$ is an integer multiple of $n$.

Can anyone point a basic textbook or online material in which this basic fact is proven?


It can be proven easily using geometric summation. Observe that $$ \sum_{k=0}^{r-1} \exp\left(i \frac{2\pi k m}{n} \right) = \sum_{k=0}^{r-1} \exp(2\pi i m/n)^k = \frac{1-\exp(2\pi i m/n)^r}{1-\exp(2\pi i m/n)} $$ The sum is $0$ if and only if $\exp(2\pi i m/n)^r = 1$, which corresponds to $n|rm$.


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