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 Properties of Gauss sums
3 See also
4 References

[edit] Definition

Let a,b,c be natural numbers. The generalized Gauss' sum G(a,b,c) is defined by

$G(a,b,c)=\sum_{n=0}^{c-1} e((a n^2+bn)/c),$

where e(x) is the exponential function exp(2πix). The classical Gauss sum is the sum $G (a, c) = G (a,0, c)$ .

[edit] Properties of Gauss sums

The Gauss sum G(a,b,c) depends only on the residue class of a,b modulo c.

Gauss sums are multiplicative, i.e. given natural numbers a, b, c and d with gcd(c,d) =1 one has

G(a,b,cd)=G(ac,b,d)G(ad,b,c).

This is a direct consequence of the Chinese remainder theorem.

One has G(a,b,c)=0 if gcd(a,c)>1 except if gcd(a,c) divides b in which case one has

$G(a,b,c)= gcd(a,c) \cdot G\left(\frac{a}{gcd(a,c)},\frac{b}{gcd(a,c)},\frac{c}{gcd(a,c)}\right)$

Thus in the evaluation of quadratic Gauss sums one may always assume gcd(a,c)=1.

Let a,b and c be integers with $ac\neq 0$ and ac+b even. One has the following analogue of quadratic reciprocity for (even more general) Gauß sums

$\sum_{n=0}^{|c|-1} e^{\pi i (a n^2+bn)/c} = |c/a|^{1/2} e^{\pi i (|ac|-b^2)/(4ac)} \sum_{n=0}^{|a|-1} e^{-\pi i (c n^2+b n)/a}.$

Define $\varepsilon_m = \begin{cases} 1 & m\equiv 1\mod 4 \\ i & m\equiv 3\mod 4 \end{cases}$ for every odd integer m.

All Gauss sums with b=0 and gcd(a,c)=1 are explicitely given by

$G(a,0,c) = \begin{cases} 0 & c\equiv 2\mod 4 \\ \varepsilon_c \sqrt{c} \left(\frac{a}{c}\right) & c\ \text{odd} \\ (1+i) \varepsilon_a^{-1} \sqrt{c} \left(\frac{c}{a}\right) & a\ \text{odd}, 4\mid c\end{cases}.$

Here $\left(\frac{a}{c}\right)$ is the Jacobi symbol. This is the famous formula of Carl Friedrich Gauß.

For b>0 the Gauss sums can easily be computed by completing the square in most cases. This fails however in some cases (for example c even and b odd) which can be computed relatively easy by other means. For example if c is odd and gcd(a,c)=1 one has

$G(a,b,c) = \varepsilon_c \sqrt{c} \cdot \left(\frac{a}{c}\right) e^{-2\pi i \psi(a) b^2/c}$

where $ψ(a)$ is some number with $4\psi(a)a \equiv 1\ \text{mod}\ c$ . As another example, if 4 divides c and b is odd and as always gcd(a,c)=1 then G(a,b,c)=0. This can, for example, be proven as follows: Because of the multiplicative property of Gauss sums we only have to show that $G (a, b,2 n) = 0$ if n>1 and a,b are odd with gcd(a,c)=1. If b is odd then $a n 2 + b n$ is even for all $0\leq n < c-1$ . By Hensel's lemma, for every q, the equation $a n 2 + b n + q = 0$ has at most two solutions in $\mathbb{Z}/2^n \mathbb{Z}$ . Because of a counting argument $a n 2 + b n$ runs through all even residue classes modulo c exactly two times. The geometric sum formula then shows that $G (a, b,2 n) = 0$ .

If c is odd and squarefree and gcd(a,c)=1 then

$G(a,0,c) = \sum_{n=0}^{c-1} \left(\frac{n}{c}\right) e^{2\pi i a n/c}.$

If c is not squarefree then the right side vanishes while the left side does not. Often the right sum is also called a quadratic Gauss sum.

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.

[edit] See also

Kummer sum

[edit] References

Ireland and Rosen (1990). A Classical Introduction to Modern Number Theory. Springer-Verlag. ISBN 0-387-97329-X.
Bruce C. Berndt, Ronald J. Evans and Kenneth S. Williams (1998). Gauss and Jacobi Sums. Wiley and Sons, Inc.. ISBN 0-471-12807-4.
Henryk Iwaniek, Emmanuel Kowalski (2004). Analytic number theory. American Mathematical Society. ISBN 0-8218-3633-1.

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