Gauss sum

Template:Short description Template:Distinguish 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, Template:Math is a group homomorphism of the additive group Template:Math into the unit circle, and Template:Math is a group homomorphism of the unit group Template:Math 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 Template:Math and Template:Math) (where Template:Mvar is the complex conjugate of Template:Mvar) involves a factorTemplate:Clarify

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

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 Template:Math or Template:Math 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 Template:Math 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 Template:Math. 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 Template:Math is the conductor of Template:Mvar and Template:Math is the primitive Dirichlet character modulo Template:Math that induces Template:Mvar, then the Gauss sum of Template:Mvar is related to that of Template:Math 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, Template:Math is non-zero precisely when Template:Math is squarefree and relatively prime to Template:Math.

Other relations between Template:Math 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 Template:Math is a Dirichlet character modulo Template:Math such that Template:Mvar and Template:Math 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 Template:Math, Template:Math, and Template:Math when Template:Mvar and Template:Math are of the same modulus (and Template:Math is primitive) is measured by the Jacobi sum Template:Math. 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
• Template:Cite book
• Template:Cite book
• Section 3.4 of Template:Citation