Kramers theorem

In mathematics, 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 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, Template:Overline) involves a factor

$G(\chi )\ /\ |G(\chi )|,$

where Template:Overline 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) = p^1/2 or ip^1/2 according as p is congruent to 1 or 3 modulo 4.

An alternate form for this Gauss sum is:

$\sum e^{\frac {2\pi ir^{2}}{p}}$

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

The general theory of Gauss sums was developed in the early nineteenth 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 R is a field of p elements and χ is nontrivial, the absolute value is p^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 N is

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

If χ is moreover primitive, then

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

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

$G(\chi )=\mu (N/N_{0})\chi _{0}(N/N_{0})G(\chi _{0})~$

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

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

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

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

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,

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

See also

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

References

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