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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gauss sum ： ウィキペディア英語版 Gauss sum In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum 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 ''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'', ) (where is the complex conjugate of χ) involves a factor :$\backslash frac$. 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: :$\backslash sum\; e^\}$ 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 :$G(\backslash chi)=\backslash sum\_^N\backslash chi(a)e^.$ If χ is moreover primitive, then :$|G(\backslash chi)|=\backslash sqrt,$ in particular, it is non-zero. More generally, if ''N''0 is the conductor of χ and χ0 is the primitive Dirichlet character modulo ''N''0 that induces χ, then the Gauss sum of χ is related to that of χ0 by :$G(\backslash chi)=\backslash mu(N/N\_0)\backslash chi\_0(N/N\_0)G(\backslash chi\_0)~$ where μ is the Möbius function. Consequently, ''G''(χ) is non-zero precisely when ''N''/''N''0 is squarefree and relatively prime to ''N''0. Other relations between ''G''(χ) and Gauss sums of other characters include :$G(\backslash overline)=\backslash chi(-1)\backslash overline,$ where is the complex conjugate Dirichlet character, and if χ′ is a Dirichlet character modulo ''N''′ such that ''N'' and ''N''′ are relatively prime, then :$G(\backslash chi\backslash chi^\backslash prime)=\backslash chi(N^\backslash prime)\backslash chi^\backslash prime(N)G(\backslash chi)G(\backslash chi^\backslash prime).$ 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, :$G(\backslash chi\backslash chi^\backslash prime)=\backslash frac.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gauss sum」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.