Gauss sum

 ( Cyclotomic Fields ) 

In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite Summation of roots of unity, typically

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

where the sum is over elements of some finite ring commutative ring , is a group homomorphism of the group scheme into the unit circle, and is a group homomorphism of the unit group into the unit circle, extended to non-unit , where it takes the value 0. Gauss sums are the analogues for finite fields of the Gamma function.B. H. Gross and N. Koblitz. Gauss sums and the p-adic Γ-function. Ann. of Math. (2), 109(3):569–581, 1979.

Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet -functions, where for a Dirichlet character the equation relating and ) (where is the complex conjugate of ) involves a factor

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

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History The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for the residue field modulo a prime number , and the Legendre symbol. In this case Gauss proved that or for 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

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

Quadratic Gauss sums are closely connected with the theory of .

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

The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups. In the case where is a field of elements and is nontrivial, the absolute value is . 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.

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Properties of Gauss sums of Dirichlet characters The Gauss sum of a Dirichlet character modulo is

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

If is also primitive, then

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

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

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

where is the Möbius function. Consequently, is non-zero precisely when is squarefree and relatively prime to .Theorem 9.10 in H. L. Montgomery, R. C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, (2006).

Other relations between and Gauss sums of other characters include

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

where is the complex conjugate Dirichlet character, and if is a Dirichlet character modulo such that and are relatively prime, then

$G\backslash left(\backslash chi\backslash chi^\backslash prime\backslash right)\; =\; \backslash chi\backslash left(N^\backslash prime\backslash right)\; \backslash chi^\backslash prime(N)\; G(\backslash chi)\; G\backslash left(\backslash chi^\backslash prime\backslash right).$

The relation among , , and when and are of the same modulus (and is primitive) is measured by the Jacobi sum . Specifically,

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

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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.

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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

• (1998). 9780471128076, Wiley. ISBN 9780471128076
• (1990). 038797329X, Springer-Verlag. 038797329X
• Section 3.4 of

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