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Suppose $h \in \mathbb{N}$, is there a known non trivial upper bound for $$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right|?$$

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up vote 4 down vote accepted

This is a partial sum of a geometric series:

$$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right| =\left| \frac{1}{n} \sum_{m=1}^n q^m \right| =\left| \frac{q}{n} \frac{1-q^n}{1-q}\right| \le \frac{2}{n\left|1-q\right|}$$

with $q:=e^{(2 \pi)^2 i h}$.

If your question is whether there is an upper bound for this independent of $h$, the answer is no -- $2 \pi h$ comes arbitrarily close to integers as $h$ increases, and hence the denominator comes arbitrarily close to $0$. For such values of $h$, the summands remain arbitrarily close to $1$, and the only bound is the trivial one $(1/n)\cdot n=1$.

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