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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 [ edit ]
For an odd prime number p and an integer a , the quadratic Gauss sum g (a ; p ) is defined as
g
(
a
;
p
)
=
∑
n
=
0
p
−
1
ζ
p
a
n
2
,
{\displaystyle g(a;p)=\sum _{n=0}^{p-1}\zeta _{p}^{an^{2}},}
where
ζ
p
{\displaystyle \zeta _{p}}
is a primitive p th root of unity , for example
ζ
p
=
exp
(
2
π
i
/
p
)
{\displaystyle \zeta _{p}=\exp(2\pi i/p)}
.
Equivalently,
g
(
a
;
p
)
=
∑
n
=
0
p
−
1
(
1
+
(
n
p
)
)
ζ
p
a
n
.
{\displaystyle g(a;p)=\sum _{n=0}^{p-1}{\big (}1+\left({\tfrac {n}{p}}\right){\big )}\,\zeta _{p}^{an}.}
For a divisible by p the expression
ζ
p
a
n
2
{\displaystyle \zeta _{p}^{an^{2}}}
evaluates to
1
{\displaystyle 1}
. Hence, we have
g
(
a
;
p
)
=
p
.
{\displaystyle g(a;p)=p.}
For a not divisible by p , this expression reduces to
g
(
a
;
p
)
=
∑
n
=
0
p
−
1
(
n
p
)
ζ
p
a
n
=
G
(
a
,
(
⋅
p
)
)
,
{\displaystyle g(a;p)=\sum _{n=0}^{p-1}\left({\tfrac {n}{p}}\right)\,\zeta _{p}^{an}=G(a,\left({\tfrac {\cdot }{p}}\right)),}
where
G
(
a
,
χ
)
=
∑
n
=
0
p
−
1
χ
(
n
)
ζ
p
a
n
{\displaystyle G(a,\chi )=\sum _{n=0}^{p-1}\chi (n )\,\zeta _{p}^{an}}
is the Gauss sum defined for any character χ modulo p .
Properties [ edit ]
The value of the Gauss sum is an algebraic integer in the p th cyclotomic field
Q
(
ζ
p
)
{\displaystyle \mathbb {Q} (\zeta _{p})}
.
The evaluation of the Gauss sum for an integer a not divisible by a prime p >2 can be reduced to the case a = 1 :
g
(
a
;
p
)
=
(
a
p
)
g
(
1
;
p
)
.
{\displaystyle g(a;p)=\left({\tfrac {a}{p}}\right)g(1;p).}
The exact value of the Gauss sum for a = 1 is given by the formula:[1]
g
(
1
;
p
)
=
∑
n
=
0
p
−
1
e
2
π
i
n
2
p
=
{
(
1
+
i
)
p
if
p
≡
0
(
mod
4
)
,
p
if
p
≡
1
(
mod
4
)
,
0
if
p
≡
2
(
mod
4
)
,
i
p
if
p
≡
3
(
mod
4
)
.
{\displaystyle 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}}}
Remark
In fact, the identity
g
(
1
;
p
)
2
=
(
−
1
p
)
p
{\displaystyle g(1;p)^{2}=\left({\tfrac {-1}{p}}\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 [ edit ]
Let a , b , c be natural numbers . The generalized quadratic 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
{\displaystyle G(a,b,c)=\sum _{n=0}^{c-1}e^{2\pi i{\frac {an^{2}+bn}{c}}}}
.
The classical quadratic Gauss sum is the sum g (a , p ) = G (a , 0, p ) .
Properties
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 has
G
(
a
,
b
,
c
d
)
=
G
(
a
c
,
b
,
d
)
G
(
a
d
,
b
,
c
)
.
{\displaystyle 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 ) = 0if gcd(a , c ) >1 except if gcd(a ,c ) divides b in which case one has
G
(
a
,
b
,
c
)
=
gcd
(
a
,
c
)
⋅
G
(
a
gcd
(
a
,
c
)
,
b
gcd
(
a
,
c
)
,
c
gcd
(
a
,
c
)
)
{\displaystyle 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 , 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 sums[2]
∑
n
=
0
|
c
|
−
1
e
π
i
a
n
2
+
b
n
c
=
|
c
a
|
1
2
e
π
i
|
a
c
|
−
b
2
4
a
c
∑
n
=
0
|
a
|
−
1
e
−
π
i
c
n
2
+
b
n
a
{\displaystyle \sum _{n=0}^{|c|-1}e^{\pi i{\frac {an^{2}+bn}{c}}}=\left|{\frac {c}{a}}\right|^{\frac {1}{2}}e^{\pi i{\frac {|ac|-b^{2}}{4ac}}}\sum _{n=0}^{|a|-1}e^{-\pi i{\frac {cn^{2}+bn}{a}}}}
.
ε
m
=
{
1
if
m
≡
1
(
mod
4
)
i
if
m
≡
3
(
mod
4
)
{\displaystyle \varepsilon _{m}={\begin{cases}1&{\text{if}}\ m\equiv 1{\pmod {4}}\\i&{\text{if}}\ m\equiv 3{\pmod {4}}\end{cases}}}
for every odd integer m . The values of Gauss sums with b = 0 and gcd(a , c ) = 1 are explicitly given by
G
(
a
,
c
)
=
G
(
a
,
0
,
c
)
=
{
0
if
c
≡
2
(
mod
4
)
ε
c
c
(
a
c
)
if
c
≡
1
(
mod
2
)
(
1
+
i
)
ε
a
−
1
c
(
c
a
)
if
c
≡
0
(
mod
4
)
.
{\displaystyle 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}}}
Here (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 has
G
(
a
,
b
,
c
)
=
ε
c
c
⋅
(
a
c
)
e
−
2
π
i
ψ
(
a
)
b
2
c
,
{\displaystyle G(a,b,c)=\varepsilon _{c}{\sqrt {c}}\cdot \left({\frac {a}{c}}\right)e^{-2\pi i{\frac {\psi (a )b^{2}}{c}}},}
where ψ (a ) is some number with 4 ψ (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 ) = 0if n >1 and a , b are odd with gcd(a , c ) = 1 . If b is odd then an 2 + bn is even for all 0 ≤ n < c − 1 . By Hensel's lemma , for every q , the equation an 2 + bn + q = 0 has at most two solutions in
Z
{\displaystyle \mathbb {Z} }
/2n
Z
{\displaystyle \mathbb {Z} }
.[clarification needed ] Because of a counting argument an 2 + bn runs through all even residue classes modulo c exactly two times. The geometric sum formula then shows that G (a , b , 2n ) = 0 .
G
(
a
,
0
,
c
)
=
∑
n
=
0
c
−
1
(
n
c
)
e
2
π
i
a
n
c
.
{\displaystyle G(a,0,c)=\sum _{n=0}^{c-1}\left({\frac {n}{c}}\right)e^{\frac {2\pi ian}{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.
G
(
n
,
p
k
)
=
p
⋅
G
(
n
,
p
k
−
2
)
{\displaystyle G\left(n,p^{k}\right)=p\cdot G\left(n,p^{k-2}\right)}
holds for k ≥ 2 and an odd prime number p , and for k ≥ 4 and p = 2 .
See also [ edit ]
References [ edit ]
^ Theorem 1.2.2 in B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums , John Wiley and Sons, (1998).
Ireland; Rosen (1990). A Classical Introduction to Modern Number Theory . Springer-Verlag. ISBN 0-387-97329-X .
Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S. (1998). Gauss and Jacobi Sums . Wiley and Sons. ISBN 0-471-12807-4 .
Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic number theory . American Mathematical Society. ISBN 0-8218-3633-1 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Quadratic_Gauss_sum&oldid=1232812811 "
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