Quadratic Gauss Sum - Definition

Definition

Let p be an odd prime number and a an integer. Then the Gauss sum mod p, g(a;p), is the following sum of the pth roots of unity:

g(a;p) =\sum_{n=0}^{p-1}e^{2{\pi}ian^2/p}=\sum_{n=0}^{p-1}\zeta_p^{an^2}, \quad \zeta_p=e^{2{\pi}i/p}.

If a is not divisible by p, an alternative expression for the Gauss sum (with the same value) is

Here is the Legendre symbol, which is a quadratic character mod p. An analogous formula with a general character χ in place of the Legendre symbol defines the Gauss sum G(χ).

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