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On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

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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 …