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.
Let p be an odd prime number and a an integer. Then the Gauss sum modulo p, g(a; p), is the following sum of the pth roots of unity:
g ( a ; p ) = ∑ n = 0 p − 1 e 2 π i a n 2 p = ∑ n = 0 p − 1 ζ p a n 2 , ζ p = e 2 π i p . If a is not divisible by p, an alternative expression for the Gauss sum (with the same value and can be done by evaluating
∑ n = 0 p − 1 ( 1 + ( n p ) ) ζ p n in two different ways) is
G ( a , χ ) = ∑ n = 0 p − 1 χ ( n ) e 2 π i a n p . Here χ = (n/p) is the Legendre symbol, which is a quadratic character modulo p. An analogous formula with a general character χ in place of the Legendre symbol defines the Gauss sum G(χ).
The value of the Gauss sum is an algebraic integer in the pth cyclotomic field ℚ(ζp).The evaluation of the Gauss sum can be reduced to the case a = 1:(Caution, this is true for odd
p.)
The exact value of the Gauss sum, computed by Gauss, is given by the formulaThe fact thatwas 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, Peter Gustav Lejeune
Dirichlet, Leopold
Kronecker, Issai
Schur and other mathematicians found different proofs.
Let a, b, c be natural numbers. The generalized Gauss sum G(a, b, c) is defined by
G ( a , b , c ) = ∑ n = 0 c − 1 e 2 π i a n 2 + b n c , The classical Gauss sum is the sum G(a, c) = G(a, 0, c).
The Gauss sum G(a,b,c) depends only on the residue class of a and b modulo c.Gauss sums are multiplicative, i.e. given natural numbers a, b, c, d with gcd(c, d) = 1 one hasThis 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 hasThus in the evaluation of quadratic Gauss sums one may always assume
gcd(a, c) = 1.
Let a, b, c be integers with ac ≠ 0 and ac + b even. One has the following analogue of the quadratic reciprocity law for (even more general) Gauss sumsDefinefor every odd integer
m. The values of Gauss sums with
b = 0 and
gcd(a, c) = 1 are explicitly given byHere
(a/c) 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 haswhere
ψ(a) is some number with
4ψ(a)a ≡ 1 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 proved as follows: Because of the multiplicative property of Gauss sums we only have to show that
G(a, b, 2n) = 0 if
n > 1 and
a, b are odd with
gcd(a, c) = 1. If
b is odd then
an2 + bn is even for all
0 ≤ n < c − 1. By
Hensel's lemma, for every
q, the equation
an2 + bn + q = 0 has at most two solutions in
ℤ/2nℤ. Because of a counting argument
an2 + bn runs through all even residue classes modulo
c exactly two times. The geometric sum formula then shows that
G(a, b, 2n) = 0.
If c is odd and squarefree and gcd(a, c) = 1 thenIf
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 isif
k ≥ 2 and
p is an odd prime number or if
k ≥ 4 and
p = 2.