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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 Dirichlet character ： ウィキペディア英語版 Dirichlet character In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of $\backslash mathbb\; Z\; /\; k\; \backslash mathbb\; Z$. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties. If $\backslash chi$ is a Dirichlet character, one defines its Dirichlet ''L''-series by :$L(s,\backslash chi)\; =\; \backslash sum\_^\backslash infty\; \backslash frac$ where ''s'' is a complex number with real part > 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. Dirichlet ''L''-functions are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis. Dirichlet characters are named in honour of Peter Gustav Lejeune Dirichlet. ==Axiomatic definition== A Dirichlet character is any function $\backslash chi$ from the integers $\backslash mathbb$ to the complex numbers $\backslash mathbb$ such that $\backslash chi$ has the following properties:〔Montgomery & Vaughan (2007) pp.117–8〕 #There exists a positive integer ''k'' such that χ(''n'') = χ(''n'' + ''k'') for all ''n''. #If gcd(''n'',''k'') > 1 then χ(''n'') = 0; if gcd(''n'',''k'') = 1 then χ(''n'') ≠ 0. #χ(''mn'') = χ(''m'')χ(''n'') for all integers ''m'' and ''n''. From this definition, several other properties can be deduced. By property 3), χ(1)=χ(1×1)=χ(1)χ(1). Since gcd(1, ''k'') = 1, property 2) says χ(1) ≠ 0, so χ(1) = 1. Properties 3) and 4) show that every Dirichlet character χ is completely multiplicative. Property 1) says that a character is periodic with period ''k''; we say that $\backslash chi$ is a character to the modulus ''k''. This is equivalent to saying that If ''a'' ≡ ''b'' (mod ''k'') then χ(''a'') = χ(''b''). If gcd(''a'',''k'') = 1, Euler's theorem says that ''a''φ(''k'') ≡ 1 (mod ''k'') (where φ(''k'') is the totient function). Therefore by 5) and 4), χ(''a''φ(''k'')) = χ(1) = 1, and by 3), χ(''a''φ(''k'')) =χ(''a'')φ(''k''). So For all ''a'' relatively prime to ''k'', χ(''a'') is a φ(''k'')-th complex root of unity. The unique character of period 1 is called the trivial character. Note that any character vanishes at 0 except the trivial one, which is 1 on all integers. A character is called principal if it assumes the value 1 for arguments coprime to its modulus and otherwise is 0.〔Montgomery & Vaughan (2007) p.115〕 A character is called real if it assumes real values only. A character which is not real is called complex.〔Montgomery & Vaughan (2007) p.123〕 The sign of the character $\backslash chi$ depends on its value at −1. Specifically, $\backslash chi$ is said to be odd if $\backslash chi\; (-1)\; =\; -1$ and even if $\backslash chi\; (-1)\; =\; 1$. 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「Dirichlet character」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.