Gaussian sum

The Gaussian sum, Gaussian sum or Gaussian sum (not to be confused with the Gaussian sum formula ) is a certain type of finite sum of roots of unity , typically

${\displaystyle G(\chi ):=G(\chi ,\psi )=\sum _{r}\chi (r)\cdot \psi (r)}$

The sum goes over the elements of a finite commutative ring , is a group homomorphism of the Abelian group in the unit circle and is a group homomorphism of the unit group in the unit circle, continued (by the value 0) to non- units . Such sums are omnipresent in number theory . You can find e.g. B. Use in the functional equations of the Dirichlet L function , where for a Dirichlet character the equation in the relationship between and the factor ${\displaystyle r}$ ${\displaystyle R}$ ${\displaystyle \psi }$ ${\displaystyle R^{+}}$ ${\displaystyle \chi }$ ${\displaystyle R^{\times }}$ ${\displaystyle \chi }$ ${\displaystyle L(s,\chi )}$ ${\displaystyle L(1-s,\chi ^{*})}$

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

used, where the complex is conjugate of . ${\displaystyle \chi ^{*}}$ ${\displaystyle \chi }$

Originally looked at Carl Friedrich Gauss , the quadratic Gaussian sum with as a residue field modulo an odd prime number and the Legendre symbol , the square residue class character mod . Gauss proved that or holds, depending on whether it is congruent to 1 or 3 modulo 4. ${\displaystyle R}$ ${\displaystyle p}$ ${\displaystyle \chi }$ ${\displaystyle p}$ ${\displaystyle G(\chi )={\sqrt {p}}}$ ${\displaystyle G(\chi )=i{\sqrt {p}}}$ ${\displaystyle p}$

An alternative form of this Gaussian sum is:

${\displaystyle \sum _{r=0}^{p-1}e^{{\frac {2\pi i}{p}}r^{2}}}$

Quadratic Gaussian sums are closely related to the theory of theta functions .

The general theory of Gaussian sums was developed in the early 19th century, using Jacobi sums and their prime number decomposition into circles . Sums over the quantities where assumes a certain value when the underlying ring is the remainder class ring modulo an integer are described by the theory of Gaussian periods . ${\displaystyle \chi }$ ${\displaystyle N}$

The absolute value of a Gaussian sum is usually used as an application of Plancherel's theorem to finite groups. In the case that a field of elements and is nontrivial, this amount is the same . The determination of the actual value of general Gaussian sums from the result of Gauss for the quadratic case is a long unsolved problem. For some cases see sorrow sum . ${\displaystyle R}$ ${\displaystyle p}$ ${\displaystyle \chi }$ ${\displaystyle {\sqrt {p}}}$

See also

• Stickelberger's theorem

credentials

• Kenneth Ireland, Michael Rosen : A Classical Introduction to Modern Number Theory (= Graduate texts in mathematics. Vol. 84). 2nd edition. Springer-Verlag, New York et al. 1990, ISBN 0-387-97329-X .
• Bruce C. Berndt , Ronald J. Evans, Kenneth S. Williams: Gauss and Jacobi Sums (= Canadian Mathematical Society series of monographs and advanced texts. Vol. 21 = A Wiley-interscience publication). Wiley, New York NY et al. 1998, ISBN 0-471-12807-4 .