Gauss sum
Encyclopedia
In mathematics
, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
where the sum is over elements r of some finite commutative ring R, ψ(r) is a group homomorphism of the additive group
R+ into the unit circle
, and χ(r) is a group homomorphism of the unit group R× into the unit circle, extended to non-unit r where it takes the value 0. Gauss sums are the analogues for finite fields of the Gamma function
.
Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet L-functions, where for a Dirichlet character
χ the equation relating L(s, χ) and L(1 − s, ) involves a factor
where is the complex conjugate of χ.
The case originally considered by C. F. Gauss was the quadratic Gauss sum
, for R the field of residues modulo a prime number p, and χ the Legendre symbol
. In this case Gauss proved that G(χ) = p1/2 or ip1/2 according as p is congruent to 1 or 3 modulo 4.
An alternate form for this Gauss sum is:
Quadratic Gauss sums are closely connected with the theory of theta-functions.
The general theory of Gauss sums was developed in the early nineteenth century, with the use of Jacobi sum
s and their prime decomposition in cyclotomic field
s. Gauss sums over a residue ring of integers mod N are linear combinations of closely related sums called Gaussian period
s.
The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups. In the case where R is a field of p elements and χ is nontrivial, the absolute value is p1/2. The determination of the exact value of general Gauss sums, following the result of Gauss on the quadratic case, is a long-standing issue. For some cases see Kummer sum
.
If χ is moreover primitive, then
in particular, it is non-zero. More generally, if N0 is the conductor of χ and χ0 is the primitive Dirichlet character modulo N0 that induces χ, then the Gauss sum of χ is related to that of χ0 by
where μ is the Möbius function
. Consequently, G(χ) is non-zero precisely when N/N0 is squarefree and relatively prime to N0.
Other relations between G(χ) and Gauss sums of other characters include
where is the complex conjugate Dirichlet character, and if χ′ is a Dirichlet character modulo N′ such that N and N′ are relatively prime, then
The relation among G(χχ′), G(χ), and G(χ′) when χ and χ′ are of the same modulus (and χχ′ is primitive) is measured by the Jacobi sum
J(χ, χ′). Specifically,
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
where the sum is over elements r of some finite commutative ring R, ψ(r) is a group homomorphism of the additive group
Additive group
An additive group may refer to:*an abelian group, when it is written using the symbol + for its binary operation*a group scheme representing the underlying-additive-group functor...
R+ into the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
, and χ(r) is a group homomorphism of the unit group R× into the unit circle, extended to non-unit r where it takes the value 0. Gauss sums are the analogues for finite fields of the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
.
Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet L-functions, where for a Dirichlet character
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
χ the equation relating L(s, χ) and L(1 − s, ) involves a factor
where is the complex conjugate of χ.
The case originally considered by C. F. Gauss was the quadratic Gauss sum
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...
, for R the field of residues modulo a prime number p, and χ the Legendre symbol
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
. In this case Gauss proved that G(χ) = p1/2 or ip1/2 according as p is congruent to 1 or 3 modulo 4.
An alternate form for this Gauss sum is:
Quadratic Gauss sums are closely connected with the theory of theta-functions.
The general theory of Gauss sums was developed in the early nineteenth century, with the use of Jacobi sum
Jacobi sum
In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J for Dirichlet characters \chi, \psi modulo a prime number p, defined by...
s and their prime decomposition in cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s. Gauss sums over a residue ring of integers mod N are linear combinations of closely related sums called Gaussian period
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
s.
The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups. In the case where R is a field of p elements and χ is nontrivial, the absolute value is p1/2. The determination of the exact value of general Gauss sums, following the result of Gauss on the quadratic case, is a long-standing issue. For some cases see Kummer sum
Kummer sum
In mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus p, with p congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers...
.
Properties of Gauss sums of Dirichlet characters
The Gauss sum of a Dirichlet character modulo N isIf χ is moreover primitive, then
in particular, it is non-zero. More generally, if N0 is the conductor of χ and χ0 is the primitive Dirichlet character modulo N0 that induces χ, then the Gauss sum of χ is related to that of χ0 by
where μ is the Möbius function
Möbius function
The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
. Consequently, G(χ) is non-zero precisely when N/N0 is squarefree and relatively prime to N0.
Other relations between G(χ) and Gauss sums of other characters include
where is the complex conjugate Dirichlet character, and if χ′ is a Dirichlet character modulo N′ such that N and N′ are relatively prime, then
The relation among G(χχ′), G(χ), and G(χ′) when χ and χ′ are of the same modulus (and χχ′ is primitive) is measured by the Jacobi sum
Jacobi sum
In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J for Dirichlet characters \chi, \psi modulo a prime number p, defined by...
J(χ, χ′). Specifically,
See also
- Stickelberger's theoremStickelberger's theoremIn mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields...
- Hasse-Davenport relation
- Chowla-Mordell theorem
- Gaussian periodGaussian periodIn mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...