Quadratic Gauss sum

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For the general type of Gauss sums see Gaussian period, Gauss sum

In mathematics, quadratic Gauss sums are certain sums over exponential functions with quadratic argument. They are named after Carl Friedrich Gauss, who studied them extensively.

1 Definition
2 Evaluation of Gauss sums
3 See also
4 References

[edit] Definition

Gauss' sum G(n,m) is defined by

where e(x) is the exponential function exp(2πix).

[edit] Evaluation of Gauss sums

For any natural numbers n and m, G(n,m) can be evaluated using the following three results:

Gauss sums are multiplicative, i.e. given natural numbers n, m and M with gcd(m,M) =1 one has

G(n,mM)=G(nM,m)G(nm,M).

If m is odd, then

<math>G(n,m)=\begin{cases}\left(\frac{n}{m}\right)\sqrt{m} & \mbox{ if }m\equiv 1\mod(4) \\ i\left(\frac{n}{m}\right)\sqrt{m} & \mbox{ if }m\equiv 3\mod(4)\end{cases}</math>

where the fraction in parentheses is the Jacobi symbol.

If m is a power of 2, then

<math>G(n,2^l)=\begin{cases}0 & \mbox{ if }l=1 \\ \left(1+i^n\right)2^{l/2} & \mbox{ if }l\mbox{ is even} \\ 2^\frac{l+1}{2}e^{\frac{\pi i}{4}n} & \mbox{ if }l>1\mbox{ and } l\mbox{ is odd}\end{cases}</math>

where, again, the fraction in parentheses is the Jacobi symbol.

Another useful formula is

G(n,p^k)=pG(n,p^k-2)

if k≥2 and p is an odd prime number or if k≥4 and p=2.