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| | She is recognized by the title of Myrtle Shryock. It's not a typical factor but what she likes performing is foundation jumping and now she is trying to make cash with it. Her family lives in Minnesota. Hiring is my profession.<br><br>my weblog: diet meal delivery, [http://5h0w.me/dietfooddelivery78594 Suggested Resource site], |
| In [[number theory]], '''Dirichlet characters''' are certain [[arithmetic function]]s which arise from [[completely multiplicative]] [[character theory|characters]] on the units of <math> \mathbb Z / k \mathbb Z </math>. Dirichlet characters are used to define [[Dirichlet L-function|Dirichlet ''L''-functions]], which are [[meromorphic function]]s with a variety of interesting analytic properties.
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| If <math>\chi</math> is a Dirichlet character, one defines its Dirichlet ''L''-series by
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| :<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}</math>
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| 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|Riemann zeta-function]] and appear prominently in the [[generalized Riemann hypothesis]].
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| Dirichlet characters are named in honour of [[Peter Gustav Lejeune Dirichlet]].
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| ==Axiomatic definition==
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| A Dirichlet character is any [[function (mathematics)|function]] <math>\chi</math> from the [[integer]]s <math> \mathbb{Z} </math> to the [[complex number]]s <math> \mathbb{C} </math> such that <math>\chi</math> has the following properties:<ref name=MV1178>Montgomery & Vaughan (2007) pp.117–8</ref>
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| #There exists a positive integer ''k'' such that χ(''n'') = χ(''n'' + ''k'') for all ''n''.
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| #If [[greatest common divisor|gcd]](''n'',''k'') > 1 then χ(''n'') = 0; if gcd(''n'',''k'') = 1 then χ(''n'') ≠ 0.
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| #χ(''mn'') = χ(''m'')χ(''n'') for all integers ''m'' and ''n''.
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| From this definition, several other properties can be deduced.
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| By property 3), χ(1)=χ(1×1)=χ(1)χ(1). Since gcd(1, ''k'') = 1, property 2) says χ(1) ≠ 0, so
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| <ol start=4><li>χ(1) = 1.</ol>
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| Properties 3) and 4) show that every Dirichlet character χ is [[completely multiplicative]].
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| Property 1) says that a character is [[periodic function|periodic]] with period ''k''; we say that <math>\chi</math> is a character to the '''modulus''' ''k''. This is equivalent to saying that
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| <ol start=5><li>If ''a'' ≡ ''b'' (mod ''k'') then χ(''a'') = χ(''b''). </ol>
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| If gcd(''a'',''k'') = 1, [[Euler's theorem]] says that ''a''<sup>φ(''k'')</sup> ≡ 1 (mod ''k'') (where φ(''k'') is the [[totient function]]). Therefore by 5) and 4), χ(''a''<sup>φ(''k'')</sup>) = χ(1) = 1, and by 3), χ(''a''<sup>φ(''k'')</sup>) =χ(''a'')<sup>φ(''k'')</sup>. So
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| <ol start=6><li>For all ''a'' relatively prime to ''k'', χ(''a'') is a φ(''k'')-th complex [[root of unity]].</ol>
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| 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.
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| A character is called '''principal''' if it assumes the value 1 for arguments coprime to its modulus and otherwise is 0.<ref name=MV115>Montgomery & Vaughan (2007) p.115</ref> A character is called '''real''' if it assumes real values only. A character which is not real is called '''complex'''.<ref name=MV123>Montgomery & Vaughan (2007) p.123</ref>
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| The '''sign''' of the character <math>\chi</math> depends on its value at −1. Specifically, <math>\chi</math> is said to be '''odd''' if <math>\chi (-1) = -1</math> and '''even''' if <math>\chi (-1) = 1</math>.
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| ==Construction via residue classes==
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| Dirichlet characters may be viewed in terms of the [[character group]] of the
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| unit group of the ring '''Z'''/''k'''''Z''', as ''extended residue class characters''.<ref>Fröhlich & Taylor (1991) p.218</ref>
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| === Residue classes ===
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| Given an integer ''k'', one defines the '''residue class''' of an integer ''n'' as the set of all integers congruent to ''n'' [[modular arithmetic|modulo]] ''k'':
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| <math>\hat{n}=\{m | m \equiv n \mod k \}.</math>
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| That is, the residue class <math>\hat{n}</math> is the [[coset]] of ''n'' in the [[quotient ring]] '''Z'''/''k'''''Z'''.
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| The set of units modulo ''k'' forms an [[abelian group]] of order <math>\phi(k)</math>, where group multiplication is given by
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| <math>\widehat{mn}=\hat{m}\hat{n}</math> and <math>\phi</math>
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| again denotes [[Euler's phi function]].
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| The identity in this group is the residue class <math>\hat{1}</math> and the inverse of <math>\hat{m}</math> is the residue class <math>\hat{n}</math> where
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| <math>\hat{m} \hat{n} = \hat{1}</math>, i.e., <math>m n \equiv 1 \mod k</math>. For example, for ''k''=6, the set of units is <math>\{\hat{1}, \hat{5}\}</math> because 0, 2, 3, and 4 are not coprime to 6.
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| The character group of ('''Z'''/''k'')<sup>*</sup> consists of the ''residue class characters''. A residue class character θ on ('''Z'''/''k'')<sup>*</sup> is '''primitive''' if there is no proper divisor ''d'' of ''k'' such that θ factors as a map ('''Z'''/''k'')<sup>*</sup> → ('''Z'''/''d'')<sup>*</sup> → '''C'''<sup>*</sup>.<ref name=FT215>Frohlich & Taylor (1991) p.215</ref>
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| ===Dirichlet characters===
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| The definition of a Dirichlet character modulo ''k'' ensures that it restricts to a [[Character group|character]] of the unit group modulo ''k'':<ref name=A139>Apostol (1976) p.139</ref> a group homomorphism <math>\chi</math> from ('''Z'''/''k'''''Z''')<sup>*</sup> to the non-zero complex numbers
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| :<math> \chi : (\mathbb{Z}/k\mathbb{Z})^* \to \mathbb{C}^* </math>,
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| with values that are necessarily roots of unity since the units modulo ''k'' form a finite group. In the opposite direction, given a group homomorphism <math>\chi</math> on the unit group modulo ''k'', we can [[Lift (mathematics)|lift]] to a [[completely multiplicative]] function on integers relatively prime to ''k'' and then extend this function to all integers by defining it to be 0 on integers having a non-trivial factor in common with ''k''. The resulting function will then be a Dirichlet character.<ref name=A138>Apostol (1976) p.138</ref> | |
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| The '''principal character''' <math>\chi_1</math> modulo ''k'' has the properties<ref name=A138/>
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| :<math>\chi_1(n)=1</math> if gcd(''n'', ''k'') = 1 and
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| :<math>\chi_1(n)=0</math> if gcd(''n'', ''k'') > 1.
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| The associated character of the multiplicative group ('''Z'''/''k'''''Z''')<sup>*</sup> is the ''principal'' character which always takes the value 1.<ref name=A134>Apostol (1976) p.134</ref>
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| When ''k'' is 1, the principal character modulo ''k'' is equal to 1 at all integers. For ''k'' greater than 1, the principal character modulo ''k'' vanishes at integers having a non-trivial common factor with ''k'' and is 1 at other integers.
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| There are φ(''n'') Dirichlet characters modulo ''n''.<ref name=A138/>
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| ==A few character tables==
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| The tables below help illustrate the nature of a Dirichlet character. They present all of the characters from modulus 1 to modulus 10. The characters χ<sub>1</sub> are the principal characters.
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| ===Modulus 1===
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| There is <math>\phi(1)=1</math> character modulo 1:
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
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| | '''0'''
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| |-
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| | <math>\chi_1(n)</math>
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| | 1
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| |-
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| |}
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| This is the trivial character.
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| ===Modulus 2===
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| There is <math>\phi(2)=1</math> character modulo 2:
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
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| | '''0'''
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| | '''1'''
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| |-
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| | <math>\chi_1(n)</math>
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| | 0
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| | 1
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| |-
| |
| |}
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| Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.
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| ===Modulus 3===
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| There are <math>\phi(3)=2</math> characters modulo 3:
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
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| | '''0'''
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| | '''1'''
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| | '''2'''
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| |-
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| | <math>\chi_1(n)</math>
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| | 0
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| | 1
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| | 1
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| |-
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| | <math>\chi_2(n)</math>
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| | 0
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| | 1
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| | −1
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| |-
| |
| |}
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| Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 3.
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| ===Modulus 4===
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| There are <math>\phi(4)=2</math> characters modulo 4:
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
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| | '''0'''
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| | '''1'''
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| | '''2'''
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| | '''3'''
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| |-
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| | <math>\chi_1(n)</math>
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| | 0
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| | 1
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| | 0
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| | 1
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| |-
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| | <math>\chi_2(n)</math>
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| | 0
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| | 1
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| | 0
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| | −1
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| |-
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| |}
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| Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 4.
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| The Dirichlet ''L''-series for <math>\chi_1(n)</math> is
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| the Dirichlet lambda function (closely related to the [[Dirichlet eta function]])
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| :<math>L(\chi_1, s)= (1-2^{-s})\zeta(s)\, </math>
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| where <math>\zeta(s)</math> is the Riemann zeta-function. The ''L''-series for <math>\chi_2(n)</math> is the [[Dirichlet beta function|Dirichlet beta-function]]
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| :<math>L(\chi_2, s)=\beta(s).\, </math>
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| ===Modulus 5===
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| There are <math>\phi(5)=4</math> characters modulo 5. In the tables, ''i'' is a square root of <math>-1</math>.
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
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| | '''0'''
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| | '''1'''
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| | '''2'''
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| | '''3'''
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| | '''4'''
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| |-
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| | <math>\chi_1(n)</math>
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| | 0
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| | 1
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| | 1
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| | 1
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| | 1
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| |-
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| | <math>\chi_2(n)</math>
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| | 0
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| | 1
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| | ''i''
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| | −i
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| | −1
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| |-
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| | <math>\chi_3(n)</math>
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| | 0
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| | 1
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| | −1
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| | −1
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| | 1
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| |-
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| | <math>\chi_4(n)</math>
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| | 0
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| | 1
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| | −''i''
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| | ''i''
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| | −1
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| |-
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| |}
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| Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 5.
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| ===Modulus 6===
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| There are <math>\phi(6)=2</math> characters modulo 6:
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
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| | '''0'''
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| | '''1'''
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| | '''2'''
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| | '''3'''
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| | '''4'''
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| | '''5'''
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| |-
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| | <math>\chi_1(n)</math>
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| | 0
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| | 1
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| | 0
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| | 0
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| | 0
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| | 1
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| |-
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| | <math>\chi_2(n)</math>
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| | 0
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| | 1
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| | 0
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| | 0
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| | 0
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| | −1
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| |-
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| |}
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| Note that χ is wholly determined by χ(5) since 5 generates the group of units modulo 6.
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| ===Modulus 7===
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| There are <math>\phi(7)=6</math> characters modulo 7. In the table below, <math>\omega = \exp( \pi i /3).</math>
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
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| | '''0'''
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| | '''1'''
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| | '''2'''
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| | '''3'''
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| | '''4'''
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| | '''5'''
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| | '''6'''
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| |-
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| | <math>\chi_1(n)</math>
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| | 0
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| | 1
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| | 1
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| | 1
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| | 1
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| | 1
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| | 1
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| |-
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| | <math>\chi_2(n)</math>
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| | 0
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| | 1
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| | ω<sup>2</sup>
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| | ω
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| | −ω
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| | −ω<sup>2</sup>
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| | −1
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| |-
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| | <math>\chi_3(n)</math>
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| | 0
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| | 1
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| | −ω
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| | ω<sup>2</sup>
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| | ω<sup>2</sup>
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| | −ω
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| | 1
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| |-
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| | <math>\chi_4(n)</math>
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| | 0
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| | 1
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| | 1
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| | −1
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| | 1
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| | −1
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| | −1
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| |-
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| | <math>\chi_5(n)</math>
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| | 0
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| | 1
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| | ω<sup>2</sup>
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| | −ω
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| | −ω
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| | ω<sup>2</sup>
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| | 1
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| |-
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| | <math>\chi_6(n)</math>
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| | 0
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| | 1
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| | −ω
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| | −ω<sup>2</sup>
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| | ω<sup>2</sup>
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| | ω
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| | −1
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| |-
| |
| |}
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| Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 7.
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| ===Modulus 8===
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| There are <math>\phi(8)=4</math> characters modulo 8.
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
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| | '''0'''
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| | '''1'''
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| | '''2'''
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| | '''3'''
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| | '''4'''
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| | '''5'''
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| | '''6'''
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| | '''7'''
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| |-
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| | <math>\chi_1(n)</math>
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| | 0
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| | 1
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| | 0
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| | 1
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| | 0
| |
| | 1
| |
| | 0
| |
| | 1
| |
| |-
| |
| | <math>\chi_2(n)</math>
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | −1
| |
| | 0
| |
| | −1
| |
| |-
| |
| | <math>\chi_3(n)</math>
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | −1
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | −1
| |
| |-
| |
| | <math>\chi_4(n)</math>
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | −1
| |
| | 0
| |
| | −1
| |
| | 0
| |
| | 1
| |
| |-
| |
| |}
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| Note that χ is wholly determined by χ(3) and χ(5) since 3 and 5 generate the group of units modulo 8.
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| ===Modulus 9===
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| There are <math>\phi(9)=6</math> characters modulo 9. In the table below, <math>\omega = \exp( \pi i /3).</math>
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| :{| class="wikitable" style="text-align:right;"
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| |-
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| | χ \ ''n''
| |
| | '''0'''
| |
| | '''1'''
| |
| | '''2'''
| |
| | '''3'''
| |
| | '''4'''
| |
| | '''5'''
| |
| | '''6'''
| |
| | '''7'''
| |
| | '''8'''
| |
| |-
| |
| | <math>\chi_1(n)</math>
| |
| | 0
| |
| | 1
| |
| | 1
| |
| | 0
| |
| | 1
| |
| | 1
| |
| | 0
| |
| | 1
| |
| | 1
| |
| |-
| |
| | <math>\chi_2(n)</math>
| |
| | 0
| |
| | 1
| |
| | ω
| |
| | 0
| |
| | ω<sup>2</sup>
| |
| | −ω<sup>2</sup>
| |
| | 0
| |
| | −ω
| |
| | −1
| |
| |-
| |
| | <math>\chi_3(n)</math>
| |
| | 0
| |
| | 1
| |
| | ω<sup>2</sup>
| |
| | 0
| |
| | −ω
| |
| | −ω
| |
| | 0
| |
| | ω<sup>2</sup>
| |
| | 1
| |
| |-
| |
| | <math>\chi_4(n)</math>
| |
| | 0
| |
| | 1
| |
| | −1
| |
| | 0
| |
| | 1
| |
| | −1
| |
| | 0
| |
| | 1
| |
| | −1
| |
| |-
| |
| | <math>\chi_5(n)</math>
| |
| | 0
| |
| | 1
| |
| | −ω
| |
| | 0
| |
| | ω<sup>2</sup>
| |
| | ω<sup>2</sup>
| |
| | 0
| |
| | −ω
| |
| | 1
| |
| |-
| |
| | <math>\chi_6(n)</math>
| |
| | 0
| |
| | 1
| |
| | −ω<sup>2</sup>
| |
| | 0
| |
| | −ω
| |
| | ω
| |
| | 0
| |
| | ω<sup>2</sup>
| |
| | −1
| |
| |-
| |
| |}
| |
| | |
| Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 9.
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| | |
| ===Modulus 10===
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| There are <math>\phi(10)=4</math> characters modulo 10.
| |
| | |
| :{| class="wikitable" style="text-align:right;"
| |
| |-
| |
| | χ \ ''n''
| |
| | '''0'''
| |
| | '''1'''
| |
| | '''2'''
| |
| | '''3'''
| |
| | '''4'''
| |
| | '''5'''
| |
| | '''6'''
| |
| | '''7'''
| |
| | '''8'''
| |
| | '''9'''
| |
| |-
| |
| | <math>\chi_1(n)</math>
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | 0
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | 1
| |
| |-
| |
| | <math>\chi_2(n)</math>
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | ''i''
| |
| | 0
| |
| | 0
| |
| | 0
| |
| | −''i''
| |
| | 0
| |
| | −1
| |
| |-
| |
| | <math>\chi_3(n)</math>
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | −1
| |
| | 0
| |
| | 0
| |
| | 0
| |
| | −1
| |
| | 0
| |
| | 1
| |
| |-
| |
| | <math>\chi_4(n)</math>
| |
| | 0
| |
| | 1
| |
| | 0
| |
| | −''i''
| |
| | 0
| |
| | 0
| |
| | 0
| |
| | ''i''
| |
| | 0
| |
| | −1
| |
| |-
| |
| |}
| |
| | |
| Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 10.
| |
| | |
| ==Examples==
| |
| | |
| If ''p'' is an odd [[prime number]], then the function
| |
| | |
| :<math>\chi(n) = \left(\frac{n}{p}\right),\ </math> where <math>\left(\frac{n}{p}\right)</math> is the [[Legendre symbol]], is a primitive Dirichlet character modulo ''p''.<ref name=MV295>Montgomery & Vaughan (2007) p.295</ref>
| |
| | |
| More generally, if ''m'' is a positive odd number, the function
| |
| | |
| :<math>\chi(n) = \left(\frac{n}{m}\right),\ </math> where <math>\left(\frac{n}{m}\right)</math> is the [[Jacobi symbol]], is a Dirichlet character modulo ''m''.<ref name=MV295/>
| |
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| These are ''quadratic characters'': in general, the primitive quadratic characters arise precisely from the [[Kronecker symbol]].<ref name=MV296>Montgomery & Vaughan (2007) p.296</ref>
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| ==Primitive characters and conductor==
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| Residues mod ''N'' give rise to residues mod ''M'', for any factor ''M'' of ''N'', by discarding some information. The effect on Dirichlet characters goes in the opposite direction: if χ is a character mod ''M'', it ''induces'' a character χ* mod ''N'' for any multiple ''N'' of ''M''. A character is '''primitive''' if it is not induced by any character of smaller modulus.<ref name=MV123>Montgomery & Vaughan (2007) p.123</ref>
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| If χ is a character mod ''n'' and ''d'' divides ''n'', then we say that the modulus ''d'' is an ''induced modulus'' for χ if ''a'' coprime to ''n'' and 1 mod ''d'' implies χ(''a'')=1:<ref name=A166>Apostol (1976) p.166</ref> equivalently, χ(''a'') = χ(''b'') whenever ''a'', ''b'' are congruent mod ''d'' and each coprime to ''n''.<ref name=A168>Apostol (1976) p.168</ref> A character is primitive if there is no smaller induced modulus.<ref name=A168/>
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| We can formalise differently this by defining characters χ<sub>1</sub> mod ''N''<sub>1</sub> and χ<sub>2</sub> mod ''N''<sub>2</sub> to be '''co-trained''' if for some modulus ''N'' such that ''N''<sub>1</sub> and ''N''<sub>2</sub> both divide ''N'' we have χ<sub>1</sub>(''n'') = χ<sub>2</sub>(''n'') for all ''n'' coprime to ''N'': that is, there is some character χ* induced by each of χ<sub>1</sub> and χ<sub>2</sub>. This is an equivalence relation on characters. A character with the smallest modulus in an equivalence class is primitive and this smallest modulus is the '''conductor''' of the characters in the class.
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| Imprimitivity of characters can lead to missing [[Euler factor]]s in their [[Dirichlet L-function|L-function]]s.
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| ==Character orthogonality==
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| The [[orthogonality relation]]s for characters of a finite group transfer to Dirichlet characters.<ref name=A140>Apostol (1976) p.140</ref> If we fix a character χ modulo ''n'' then the sum
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| :<math>\sum_{a \bmod n} \chi(a) = 0 \ </math> | |
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| unless χ is principal, in which case the sum is φ(''n''). Similarly, if we fix a residue class ''a'' modulo ''n'' and sum over all characters we have
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| :<math> \sum_{\chi} \chi(a) = 0 \ </math>
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| unless ''a''=1 in which case the sum is φ(''n''). We deduce that any periodic function with period ''n'' supported on the residue classes prime to ''n'' is a linear combination of Dirichlet characters.<ref>Davenport (1967) pp.31–32</ref>
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| == History ==
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| Dirichlet characters and their ''L''-series were introduced by [[Peter Gustav Lejeune Dirichlet]], in 1831, in order to prove [[Dirichlet's theorem on arithmetic progressions]]. He only studied them for real ''s'' and especially as ''s'' tends to 1. The extension of these functions to complex ''s'' in the whole complex plane was obtained by [[Bernhard Riemann]] in 1859.
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| ==See also==
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| * [[Hecke character]] (also known as grössencharacter)
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| * [[Character sum]]
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| * [[Gaussian sum]]
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| * [[Primitive root modulo n|Primitive root modulo ''n'']]
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| * [[Selberg class]]
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| ==References==
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| {{reflist}}
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| * See chapter 6 of {{Apostol IANT}}
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| * {{Cite journal |doi=10.2307/2317522 |first=T. M. |last=Apostol |authorlink=Tom M. Apostol |title=Some properties of completely multiplicative arithmetical functions |journal=The American Mathematical Monthly |volume=78 |issue=3 |year=1971 |pages=266–271 |mr=0279053 | zbl=0209.34302 |jstor=2317522 }}
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| * {{cite book | last=Davenport | first=Harold | authorlink=Harold Davenport | title=Multiplicative number theory | publisher=Markham | series=Lectures in advanced mathematics | volume=1 | location=Chicago | year=1967 | zbl=0159.06303 }}
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| * {{Cite book
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| |first=Helmut
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| |last=Hasse
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| |authorlink=Helmut Hasse
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| |title=Vorlesungen über Zahlentheorie
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| |edition=2nd revised
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| |series=Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen
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| |volume=59
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| |publisher=[[Springer-Verlag]]
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| |year=1964
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| |mr=0188128 | zbl=0123.04201
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| }} see chapter 13.
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| * {{Cite arxiv |first1=R. J. |last1=Mathar |eprint=1008.2547 |class=math.NT |title=Table of Dirichlet L-series and prime zeta modulo functions for small moduli |year=2010 }}
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| * {{cite book | last1=Montgomery | first1=Hugh L | author1-link=Hugh Montgomery (mathematician) | last2=Vaughan | first2=Robert C. | author2-link=Bob Vaughan | title=Multiplicative number theory. I. Classical theory | series=Cambridge Studies in Advanced Mathematics | volume=97 | publisher=[[Cambridge University Press ]] | year=2007 | isbn=0-521-84903-9 | zbl=1142.11001 }}
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| * {{Cite journal
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| |first1=Robert
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| |last1=Spira
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| |title=Calculation of Dirichlet L-Functions
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| |journal=Mathematics of Computation
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| |volume=23
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| |pages=489–497
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| |year=1969
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| |doi=10.1090/S0025-5718-1969-0247742-X
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| |mr=0247742 | zbl=0182.07001
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| |issue=107
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| }}
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| * {{cite book | last1=Fröhlich | first1=A. | author1-link=Albrecht Fröhlich | last2=Taylor | first2= M.J. | author2-link=Martin J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-36664-X | zbl=0744.11001 }}
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| ==External links==
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| * {{springer|title=Dirichlet character|id=p/d032810}}
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| [[Category:Zeta and L-functions]]
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| [[de:Charakter (Mathematik)#Dirichlet-Charaktere]]
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