quadratic Gauss sum

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.

Definition

For an odd prime number and an integer , the quadratic Gauss sum is defined as
: $g(a;p) = \sum_^\zeta_p^,$
where $\zeta_p$ is a primitive th root of unity, for example $\zeta_p=\exp(2\pi i/p)$.
Equivalently,
: $g(a;p) = \sum_^\big(1+\left(\tfrac\right)\big)\,\zeta_p^.$

For divisible by , and we have $\zeta_p^=1$ and thus
: $g(a;p) = p.$

For not divisible by , we have $\sum_^ \zeta_p^ = 0$, implying that
: $g(a;p) = \sum_^\left(\tfrac\right)\,\zeta_p^ = G(a,\left(\tfrac\right)),$
where
: $G(a,\chi)=\sum_^\chi(n)\,\zeta_p^$
is the Gauss sum defined for any character modulo .

Properties

* The value of the Gauss sum is an algebraic integer in the th cyclotomic field $\mathbb(\zeta_p)$.
* The evaluation of the Gauss sum for an integer not divisible by a prime can be reduced to the case :
:: $g(a;p)=\left(\tfrac\right)g(1;p).$

* The exact value of the Gauss sum for is given by the formula:
:: $g(1;p) =\sum_^e^\frac= \begin (1+i)\sqrt & \text\ p\equiv 0 \pmod 4, \\ \sqrt & \text\ p\equiv 1\pmod 4, \\ 0 & \text\ p \equiv 2 \pmod 4, \\ i\sqrt & \text\ p\equiv 3\pmod 4. \end$

; Remark
In fact, the identity
: $g(1;p)^2=\left(\tfrac\right)p$
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 sums

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

:$G(a,b,c)=\sum_^ e^$.

The classical quadratic Gauss sum is the sum .

; Properties

*The Gauss sum depends only on the residue class of and modulo .
*Gauss sums are multiplicative, i.e. given natural numbers with 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 if except if divides in which case one has
::$G(a,b,c)= \gcd(a,c) \cdot G\left(\frac,\frac,\frac\right)$.

:Thus in the evaluation of quadratic Gauss sums one may always assume .

*Let be integers with and even. One has the following analogue of the quadratic reciprocity law for (even more general) Gauss sumsTheorem 1.2.2 in B. C. Berndt, R. J. Evans, K. S. Williams, ''Gauss and Jacobi Sums'', John Wiley and Sons, (1998).
::$\sum_^ e^ = \left, \frac\^\frac12 e^ \sum_^ e^$.

*Define
::$\varepsilon_m = \begin 1 & \text\ m\equiv 1\pmod 4 \\ i & \text\ m\equiv 3\pmod 4 \end$
:for every odd integer . The values of Gauss sums with and are explicitly given by

::$G(a,c) = G(a,0,c) = \begin 0 & \text\ c\equiv 2\pmod 4 \\ \varepsilon_c \sqrt \left(\dfrac\right) & \text\ c\equiv 1\pmod 2 \\ (1+i) \varepsilon_a^ \sqrt \left(\dfrac\right) & \text\ c\equiv 0\pmod 4. \end$

:Here is the Jacobi symbol. This is the famous formula of Carl Friedrich Gauss.

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

::$G(a,b,c) = \varepsilon_c \sqrt \cdot \left(\frac\right) e^,$

:where is some number with . As another example, if 4 divides and is odd and as always then . This can, for example, be proved as follows: because of the multiplicative property of Gauss sums we only have to show that if and are odd with . If is odd then is even for all . For every , the equation has at most two solutions in . Indeed, if $n_1$ and $n_2$ are two solutions of same parity, then $(n_1 - n_2)(a(n_1 + n_2) +b) = \alpha 2^m$ for some integer $\alpha$, but $(a(j_1 + j_2) +b)$ is odd, hence $j_1 \equiv j_2 \pmod$. Because of a counting argument runs through all even residue classes modulo exactly two times. The geometric sum formula then shows that .

*If is an odd square-free integer and , then

::$G(a,0,c) = \sum_^ \left(\frac\right) e^\frac.$

:If 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

::$G\left(n,p^k\right) = p\cdot G\left(n,p^\right)$

:holds for and an odd prime number , and for and .

See also

*Gauss sum
*Gaussian period
* Kummer sum
* Landsberg–Schaar relation

References

*
*
*{{cite book , first1 = Henryk , last1 = Iwaniec , first2 = Emmanuel , last2 = Kowalski , title = Analytic number theory , publisher = American Mathematical Society , year = 2004 , isbn=0-8218-3633-1

Cyclotomic fields