Incubator escapee wiki

Quadratic Gauss sum

Article Discussion

In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws.

DefinitionEdit

For an odd prime number Template:Mvar and an integer Template:Mvar, the quadratic Gauss sum Template:Math is defined as

<math> g(a;p) = \sum_{n=0}^{p-1}\zeta_p^{an^2},</math>

where <math>\zeta_p</math> is a [[Primitive nth root of unity|primitive Template:Mvarth root of unity]], for example <math>\zeta_p=\exp(2\pi i/p)</math>. Equivalently,

<math>g(a;p) = \sum_{n=0}^{p-1}\big(1+\left(\tfrac{n}{p}\right)\big)\,\zeta_p^{an}.</math>

For Template:Mvar divisible by Template:Mvar, and we have <math>\zeta_p^{an^2}=1</math> and thus

<math> g(a;p) = p.</math>

For Template:Mvar not divisible by Template:Mvar, we have <math>\sum_{n=0}^{p-1} \zeta_p^{an} = 0</math>, implying that

<math>g(a;p) = \sum_{n=0}^{p-1}\left(\tfrac{n}{p}\right)\,\zeta_p^{an} = G(a,\left(\tfrac{\cdot}{p}\right)),</math>

where

<math>G(a,\chi)=\sum_{n=0}^{p-1}\chi(n)\,\zeta_p^{an}</math>

is the Gauss sum defined for any character Template:Mvar modulo Template:Mvar.

PropertiesEdit

<math> g(a;p)=\left(\tfrac{a}{p}\right)g(1;p). </math>
<math> g(1;p) =\sum_{n=0}^{p-1}e^\frac{2\pi in^2}{p}=

\begin{cases} (1+i)\sqrt{p} & \text{if}\ p\equiv 0 \pmod 4, \\ \sqrt{p} & \text{if}\ p\equiv 1\pmod 4, \\ 0 & \text{if}\ p \equiv 2 \pmod 4, \\ i\sqrt{p} & \text{if}\ p\equiv 3\pmod 4. \end{cases}</math>

Remark

In fact, the identity

<math>g(1;p)^2=\left(\tfrac{-1}{p}\right)p</math>

was easy to prove and led to one of Gauss's proofs of quadratic reciprocity. However, the determination of the sign of the Gauss sum turned out to be considerably more difficult: Gauss could only establish it after several years' work. Later, Dirichlet, Kronecker, Schur and other mathematicians found different proofs.

Generalized quadratic Gauss sumsEdit

Let Template:Math be natural numbers. The generalized quadratic Gauss sum Template:Math is defined by

<math>G(a,b,c)=\sum_{n=0}^{c-1} e^{2\pi i\frac{a n^2+bn}{c}}</math>.

The classical quadratic Gauss sum is the sum Template:Math.

Properties
<math>G(a,b,cd)=G(ac,b,d)G(ad,b,c).</math>
This is a direct consequence of the Chinese remainder theorem.
<math>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)</math>.
Thus in the evaluation of quadratic Gauss sums one may always assume Template:Math.
<math>\sum_{n=0}^{|c|-1} e^{\pi i \frac{a n^2+bn}{c}} = \left|\frac{c}{a}\right|^\frac12 e^{\pi i \frac{|ac|-b^2}{4ac}} \sum_{n=0}^{|a|-1} e^{-\pi i \frac{c n^2+b n}{a}}</math>.
  • Define
<math> \varepsilon_m = \begin{cases} 1 & \text{if}\ m\equiv 1\pmod 4 \\ i & \text{if}\ m\equiv 3\pmod 4 \end{cases}</math>
for every odd integer Template:Math. The values of Gauss sums with Template:Math and Template:Math are explicitly given by
<math>G(a,c) = G(a,0,c) =

\begin{cases} 0 & \text{if}\ c\equiv 2\pmod 4 \\ \varepsilon_c \sqrt{c} \left(\dfrac{a}{c}\right) & \text{if}\ c\equiv 1\pmod 2 \\ (1+i) \varepsilon_a^{-1} \sqrt{c} \left(\dfrac{c}{a}\right) & \text{if}\ c\equiv 0\pmod 4. \end{cases}</math>

Here Template:Math is the Jacobi symbol. This is the famous formula of Carl Friedrich Gauss.
<math>G(a,b,c) = \varepsilon_c \sqrt{c} \cdot \left(\frac{a}{c}\right) e^{-2\pi i \frac{\psi(a) b^2}{c}},</math>
where Template:Math is some number with Template:Math. As another example, if 4 divides Template:Mvar and Template:Mvar is odd and as always Template:Math then Template:Math. This can, for example, be proved as follows: because of the multiplicative property of Gauss sums we only have to show that Template:Math if Template:Math and Template:Math are odd with Template:Math. If Template:Mvar is odd then Template:Math is even for all Template:Math. For every Template:Mvar, the equation Template:Math has at most two solutions in Template:Math. Indeed, if <math>n_1</math> and <math>n_2</math> are two solutions of same parity, then <math>(n_1 - n_2)(a(n_1 + n_2) +b) = \alpha 2^m</math> for some integer <math>\alpha</math>, but <math>(a(j_1 + j_2) +b)</math> is odd, hence <math>j_1 \equiv j_2 \pmod{2^m}</math>. Template:Huh Because of a counting argument Template:Math runs through all even residue classes modulo Template:Mvar exactly two times. The geometric sum formula then shows that Template:Math.
<math>G(a,0,c) = \sum_{n=0}^{c-1} \left(\frac{n}{c}\right) e^\frac{2\pi i a n}{c}.</math>
If Template:Mvar 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
<math>G\left(n,p^k\right) = p\cdot G\left(n,p^{k-2}\right)</math>
holds for Template:Math and an odd prime number Template:Mvar, and for Template:Math and Template:Math.

See alsoEdit

ReferencesEdit

Template:Reflist

Retrieved from "https://zalansite.site/mediawiki/index.php?title=Quadratic_Gauss_sum&oldid=360719"