Gauss sums - определение. Что такое Gauss sums
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Что (кто) такое Gauss sums - определение

SUM
Gauss sums; Gaussian sum

Gauss sum         
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
Gauss notation         
NOTATION FOR MATHEMATICAL KNOTS
Gauss code; Extended Gauss code; Extended Gauss notation; Gauss word; Gauss diagram
Gauss notation (also known as a Gauss code or Gauss word) is a notation for mathematical knots. It is created by enumerating and classifying the crossings of an embedding of the knot in a plane.
Carl Friedrich Gauss         
  • heliotrope]] (background: mathematical signs) and a section of the [[triangulation network]]
  • German 10-[[Deutsche Mark]] [[Banknote]] (1993; discontinued) with formula and graph of normal distribution (background: some Göttingen buildings); portrait as mirror image of the Jensen portrait
  • Lithography by [[Siegfried Bendixen]] (1828)
  • Brunswick]]
  • House of birth in Brunswick (destroyed in World War II)
  • German Research Centre for Geosciences]] in [[Potsdam]]
  • Gauss on his deathbed (1855)
  • [[Copley Medal]] for Gauss (1838)
  • Caricature of Abraham Gotthelf Kästner by Gauss (1795)
  • Carl Friedrich Gauß 1803 by Johann Christian August Schwartz
  • Title page of Gauss' magnum opus, ''[[Disquisitiones Arithmeticae]]''
  • [[Gauss's diary]] entry related to sum of triangular numbers (1796)
  • Portrait of Gauss in Volume II of "''Carl Friedrich Gauss Werke''," 1876
  • Title page of ''Intensitas vis Magneticae Terrestris ad Mensuram Absolutam Revocata''
  • Title page of ''Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium''
  • Title page to the English Translation of ''Theoria Motus'' by [[Charles Henry Davis]] (1857)
  • Parochial registration]] of Gauss' birth
  • [[Survey marker]] stone in Garlste (now [[Garlstedt]])
  • Old observatory (circa 1800)
  • Albani Cemetery]] in [[Göttingen]], Germany
  • Gauss-Weber monument in Göttingen
  • Gauss' second wife Wilhelmine Waldeck
  • Ludwig Becker]]
GERMAN MATHEMATICIAN AND PHYSICIST (1777–1855)
Johann Carl Friedrich Gauss; Karl Gauss; Carl Frederich Gauss; Karl Friedrich Gauss; Carl Gauss; C. F. Gauss; Carl F. Gauss; Carl Friedrich Gauß; Johann Friedrich Karl Gauss; C.F. Gauss; Carl friedrich gauss; Carl Friederich Gauss; C. F. Gauß; Guass; CF Gauss; Karl Friedrich Gauß; Carl Freidrich Gauss; Johann Carl Friedrich Gauß; Carl Gauß; Friedrich gauss; Gauss; Johann Karl Friedrich Gauss; Carolus Fridericus Gauss; Princeps mathematicorum; Religious views of Carl Friedrich Gauss; Gauß, Johann Carl Friedrich; Carl Friedrich Gausz
<person> A German mathematician (1777 - 1855), one of all time greatest. Gauss discovered the method of least squares and Gaussian elimination. Gauss was something of a child prodigy; the most commonly told story relates that when he was 10 his teacher, wanting a rest, told his class to add up all the numbers from 1 to 100. Gauss did it in seconds, having noticed that 1+...+100 = 100+...+1 = (101+...+101)/2. He did important work in almost every area of mathematics. Such eclecticism is probably impossible today, since further progress in most areas of mathematics requires much hard background study. Some idea of the range of his work can be obtained by noting the many mathematical terms with "Gauss" in their names. E.g. Gaussian elimination (linear algebra); Gaussian primes (number theory); Gaussian distribution (statistics); Gauss [unit] (electromagnetism); Gaussian curvature (differential geometry); Gaussian quadrature (numerical analysis); Gauss-Bonnet formula (differential geometry); {Gauss's identity} (hypergeometric functions); Gauss sums ({number theory}). His favourite area of mathematics was number theory. He conjectured the Prime Number Theorem, pioneered the {theory of quadratic forms}, proved the {quadratic reciprocity theorem}, and much more. He was "the first mathematician to use complex numbers in a really confident and scientific way" (Hardy & Wright, chapter 12). He nearly went into architecture rather than mathematics; what decided him on mathematics was his proof, at age 18, of the startling theorem that a regular N-sided polygon can be constructed with ruler and compasses if and only if N is a power of 2 times a product of distinct Fermat primes. (1995-04-10)

Википедия

Gauss sum

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

G ( χ ) := G ( χ , ψ ) = χ ( r ) ψ ( r ) {\displaystyle 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 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 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, χ) (where χ is the complex conjugate of χ) involves a factor

G ( χ ) | G ( χ ) | . {\displaystyle {\frac {G(\chi )}{|G(\chi )|}}.}