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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Upper bound of signed exponential sums

I am wondering whether I can get the upper bound in closed form of $$\sum_{n=1}^N(\alpha \exp(j2\pi n/N)) \text{ where } \alpha = +1\text{ or }-1 \text{ and } j^2=-1.$$ If alpha is just positive one …
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