Gauss sum

Template:Short description In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically

$G(\chi ):=G(\chi ,\psi )=\sum \chi (r)\cdot \psi (r)$

where the sum is over elements Template:Mvar of some finite commutative ring Template:Mvar, ψ is a group homomorphism of the additive group R⁺ into the unit circle, and χ is a group homomorphism of the unit group R^× into the unit circle, extended to non-unit Template:Mvar, 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-function|Dirichlet Template:Mvar-function]]s, where for a Dirichlet character Template:Mvar the equation relating L(s, χ) and L(1 − s, χ) (where Template:Mvar is the complex conjugate of Template:Mvar) involves a factor

$\frac{G(\chi )}{|G(\chi )|}.$

History

The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for Template:Mvar the field of residues modulo a prime number Template:Mvar, and Template:Mvar the Legendre symbol. In this case Gauss proved that G(χ) = p^Template:1/2 or ip^Template:1/2 for Template:Mvar congruent to 1 or 3 modulo 4 respectively (the quadratic Gauss sum can also be evaluated by Fourier analysis as well as by contour integration).

An alternate form for this Gauss sum is

$\sum e^{2\pi i{r}^2/p}$.

Quadratic Gauss sums are closely connected with the theory of theta functions.

The general theory of Gauss sums was developed in the early 19th century, with the use of Jacobi sums and their prime decomposition in cyclotomic fields. Gauss sums over a residue ring of integers mod N are linear combinations of closely related sums called Gaussian periods.

The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups. In the case where Template:Mvar is a field of Template:Mvar elements and Template:Mvar is nontrivial, the absolute value is p^Template:1/2. 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.

Properties of Gauss sums of Dirichlet characters

The Gauss sum of a Dirichlet character modulo Template:Mvar is

$G(\chi )=\sum _{a=1}^N\chi (a)e^{2\pi ia/N}.$

If Template:Mvar is also primitive, then

$|G(\chi )|=\sqrt{N},$

in particular, it is nonzero. More generally, if N₀ is the conductor of Template:Mvar and χ₀ is the primitive Dirichlet character modulo N₀ that induces Template:Mvar, then the Gauss sum of Template:Mvar is related to that of χ₀ by

$G(\chi )=\mu \left(\frac{N}{N_0}\right){\chi }_0\left(\frac{N}{N_0}\right)G\left({\chi }_0\right)$

where Template:Mvar is the Möbius function. Consequently, G(χ) is non-zero precisely when Template:Sfrac is squarefree and relatively prime to N₀.

Other relations between G(χ) and Gauss sums of other characters include

$G(\bar{\chi })=\chi (-1)\overline{G(\chi )},$

where Template:Mvar is the complex conjugate Dirichlet character, and if χ′ is a Dirichlet character modulo N′ such that Template:Mvar and N′ are relatively prime, then

$G\left(\chi {\chi }^{\prime }\right)=\chi \left(N^{\prime }\right){\chi }^{\prime }(N)G(\chi )G\left({\chi }^{\prime }\right).$

The relation among G(χχ′), G(χ), and G(χ′) when Template:Mvar and χ′ are of the same modulus (and χχ′ is primitive) is measured by the Jacobi sum J(χ, χ′). Specifically,

$G\left(\chi {\chi }^{\prime }\right)=\frac{G(\chi )G\left({\chi }^{\prime }\right)}{J(\chi ,{\chi }^{\prime })}.$

Further properties

• Gauss sums can be used to prove quadratic reciprocity, cubic reciprocity, and quartic reciprocity.
• Gauss sums can be used to calculate the number of solutions of polynomial equations over finite fields, and thus can be used to calculate certain zeta functions.

See also

• Quadratic Gauss sum
• Elliptic Gauss sum
• Jacobi sum
• Kummer sum
• Kloosterman sum
• Gaussian period
• Hasse–Davenport relation
• Chowla–Mordell theorem
• Stickelberger's theorem

References

• Template:Apostol IANT
• Section 3.4 of