Exponential Sum - Example: The Quadratic Gauss Sum

Example: The Quadratic Gauss Sum

Let p be an odd prime and let . Then the quadratic Gauss sum is given by

\sum_{n=0}^{p-1}\xi^{n^2} = \begin{cases} \sqrt{p}, & p = 1 \mod 4 \\ i\sqrt{p}, & p = 3 \mod 4 \end{cases}

where the square roots are taken to be positive.

This is the ideal degree of cancellation one could hope for without any a priori knowledge of the structure of the sum, since it matches the scaling of a random walk.

Read more about this topic:  Exponential Sum

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