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2014-01-30T03:15:37Z

80.219.115.66: /* Other examples */

{{no footnotes|date=October 2010}}
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.
If <math>\chi</math> is a Dirichlet character, one defines its Dirichlet ''L''-series by

:<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}</math>

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]].

Dirichlet characters are named in honour of [[Peter Gustav Lejeune Dirichlet]].

==Axiomatic definition==
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>

#There exists a positive integer ''k'' such that χ(''n'') = χ(''n'' + ''k'') for all ''n''.
#If [[greatest common divisor|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
<ol start=4><li>χ(1) = 1.</ol>
Properties 3) and 4) show that every Dirichlet character χ is [[completely multiplicative]].

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
<ol start=5><li>If ''a'' ≡ ''b'' (mod ''k'') then χ(''a'') = χ(''b''). </ol>
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
<ol start=6><li>For all ''a'' relatively prime to ''k'', χ(''a'') is a φ(''k'')-th complex [[root of unity]].</ol>

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.<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>

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>.

==Construction via residue classes==

Dirichlet characters may be viewed in terms of the [[character group]] of the
unit group of the ring '''Z'''/''k'''''Z''', as ''extended residue class characters''.<ref>Fröhlich & Taylor (1991) p.218</ref>

=== Residue classes ===
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'':
<math>\hat{n}=\{m | m \equiv n \mod k \}.</math>
That is, the residue class <math>\hat{n}</math> is the [[coset]] of ''n'' in the [[quotient ring]] '''Z'''/''k'''''Z'''.

The set of units modulo ''k'' forms an [[abelian group]] of order <math>\phi(k)</math>, where group multiplication is given by
<math>\widehat{mn}=\hat{m}\hat{n}</math> and <math>\phi</math>
again denotes [[Euler's phi function]].
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
<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.

The character group of ('''Z'''/''k'')* consists of the ''residue class characters''. A residue class character θ on ('''Z'''/''k'')* is '''primitive''' if there is no proper divisor ''d'' of ''k'' such that θ factors as a map ('''Z'''/''k'')* → ('''Z'''/''d'')* → '''C'''*.<ref name=FT215>Frohlich & Taylor (1991) p.215</ref>

===Dirichlet characters===
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''')* to the non-zero complex numbers

:<math> \chi : (\mathbb{Z}/k\mathbb{Z})^* \to \mathbb{C}^* </math>,

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>

The '''principal character''' <math>\chi_1</math> modulo ''k'' has the properties<ref name=A138/>

:<math>\chi_1(n)=1</math> if gcd(''n'', ''k'') = 1 and
:<math>\chi_1(n)=0</math> if gcd(''n'', ''k'') > 1.

The associated character of the multiplicative group ('''Z'''/''k'''''Z''')* is the ''principal'' character which always takes the value 1.<ref name=A134>Apostol (1976) p.134</ref>

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.

There are φ(''n'') Dirichlet characters modulo ''n''.<ref name=A138/>

==A few character tables==
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 χ1 are the principal characters.

===Modulus 1===
There is <math>\phi(1)=1</math> character modulo 1:

:{| class="wikitable" style="text-align:right;"
|-
| χ \ ''n''  
|   '''0'''  
|-
| <math>\chi_1(n)</math>
| 1
|-
|}

This is the trivial character.

===Modulus 2===
There is <math>\phi(2)=1</math> character modulo 2:

:{| class="wikitable" style="text-align:right;"
|-
| χ \ ''n''  
|   '''0'''  
|   '''1'''  
|-
| <math>\chi_1(n)</math>
| 0
| 1
|-
|}

Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.

===Modulus 3===
There are <math>\phi(3)=2</math> characters modulo 3:

:{| class="wikitable" style="text-align:right;"
|-
| χ \ ''n''  
|   '''0'''  
|   '''1'''  
|   '''2'''  
|-
| <math>\chi_1(n)</math>
| 0
| 1
| 1
|-
| <math>\chi_2(n)</math>
| 0
| 1
| −1
|-
|}

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 3.

===Modulus 4===
There are <math>\phi(4)=2</math> characters modulo 4:

:{| class="wikitable" style="text-align:right;"
|-
| χ \ ''n''  
|   '''0'''  
|   '''1'''  
|   '''2'''  
|   '''3'''  
|-
| <math>\chi_1(n)</math>
| 0
| 1
| 0
| 1
|-
| <math>\chi_2(n)</math>
| 0
| 1
| 0
| −1
|-
|}

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 4.

The Dirichlet ''L''-series for <math>\chi_1(n)</math> is
the Dirichlet lambda function (closely related to the [[Dirichlet eta function]])

:<math>L(\chi_1, s)= (1-2^{-s})\zeta(s)\, </math>

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]]

:<math>L(\chi_2, s)=\beta(s).\, </math>

===Modulus 5===
There are <math>\phi(5)=4</math> characters modulo 5. In the tables, ''i'' is a square root of <math>-1</math>.

:{| class="wikitable" style="text-align:right;"
|-
| χ \ ''n''  
|   '''0'''  
|   '''1'''  
|   '''2'''  
|   '''3'''  
|   '''4'''  
|-
| <math>\chi_1(n)</math>
| 0
| 1
| 1
| 1
| 1
|-
| <math>\chi_2(n)</math>
| 0
| 1
| ''i''
| −i
| −1
|-
| <math>\chi_3(n)</math>
| 0
| 1
| −1
| −1
| 1
|-
| <math>\chi_4(n)</math>
| 0
| 1
| −''i''
| ''i''
| −1
|-
|}

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 5.

===Modulus 6===
There are <math>\phi(6)=2</math> characters modulo 6:

:{| class="wikitable" style="text-align:right;"
|-
| χ \ ''n''  
|   '''0'''  
|   '''1'''  
|   '''2'''  
|   '''3'''  
|   '''4'''  
|   '''5'''  
|-
| <math>\chi_1(n)</math>
| 0
| 1
| 0
| 0
| 0
| 1
|-
| <math>\chi_2(n)</math>
| 0
| 1
| 0
| 0
| 0
| −1
|-
|}

Note that χ is wholly determined by χ(5) since 5 generates the group of units modulo 6.

===Modulus 7===
There are <math>\phi(7)=6</math> characters modulo 7. In the table below, <math>\omega = \exp( \pi i /3).</math>

:{| class="wikitable" style="text-align:right;"
|-
| χ \ ''n''  
|   '''0'''  
|   '''1'''  
|   '''2'''  
|   '''3'''  
|   '''4'''  
|   '''5'''  
|   '''6'''  
|-
| <math>\chi_1(n)</math>
| 0
| 1
| 1
| 1
| 1
| 1
| 1
|-
| <math>\chi_2(n)</math>
| 0
| 1
| ω2
| ω
| −ω
| −ω2
| −1
|-
| <math>\chi_3(n)</math>
| 0
| 1
| −ω
| ω2
| ω2
| −ω
| 1
|-
| <math>\chi_4(n)</math>
| 0
| 1
| 1
| −1
| 1
| −1
| −1
|-
| <math>\chi_5(n)</math>
| 0
| 1
| ω2
| −ω
| −ω
| ω2
| 1
|-
| <math>\chi_6(n)</math>
| 0
| 1
| −ω
| −ω2
| ω2
| ω
| −1
|-
|}

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 7.

===Modulus 8===
There are <math>\phi(8)=4</math> characters modulo 8.

:{| class="wikitable" style="text-align:right;"
|-
| χ \ ''n''  
|   '''0'''  
|   '''1'''  
|   '''2'''  
|   '''3'''  
|   '''4'''  
|   '''5'''  
|   '''6'''  
|   '''7'''  
|-
| <math>\chi_1(n)</math>
| 0
| 1
| 0
| 1
| 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
|-
|}

Note that χ is wholly determined by χ(3) and χ(5) since 3 and 5 generate the group of units modulo 8.

===Modulus 9===
There are <math>\phi(9)=6</math> characters modulo 9. In the table below, <math>\omega = \exp( \pi i /3).</math>

:{| class="wikitable" style="text-align:right;"
|-
| 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
| ω2
| −ω2
| 0
| −ω
| −1
|-
| <math>\chi_3(n)</math>
| 0
| 1
| ω2
| 0
| −ω
| −ω
| 0
| ω2
| 1
|-
| <math>\chi_4(n)</math>
| 0
| 1
| −1
| 0
| 1
| −1
| 0
| 1
| −1
|-
| <math>\chi_5(n)</math>
| 0
| 1
| −ω
| 0
| ω2
| ω2
| 0
| −ω
| 1
|-
| <math>\chi_6(n)</math>
| 0
| 1
| −ω2
| 0
| −ω
| ω
| 0
| ω2
| −1
|-
|}

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 9.

===Modulus 10===
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/>

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>

==Primitive characters and conductor==
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>

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/>

We can formalise differently this by defining characters χ1 mod ''N''1 and χ2 mod ''N''2 to be '''co-trained''' if for some modulus ''N'' such that ''N''1 and ''N''2 both divide ''N'' we have χ1(''n'') = χ2(''n'') for all ''n'' coprime to ''N'': that is, there is some character χ* induced by each of χ1 and χ2. 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.

Imprimitivity of characters can lead to missing [[Euler factor]]s in their [[Dirichlet L-function|L-function]]s.

==Character orthogonality==
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

:<math>\sum_{a \bmod n} \chi(a) = 0 \ </math>

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

:<math> \sum_{\chi} \chi(a) = 0 \ </math>

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>

== History ==
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.

==See also==
* [[Hecke character]] (also known as grössencharacter)
* [[Character sum]]
* [[Gaussian sum]]
* [[Primitive root modulo n|Primitive root modulo ''n'']]
* [[Selberg class]]

==References==
{{reflist}}
* See chapter 6 of {{Apostol IANT}}
* {{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 }}
* {{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 }}
* {{Cite book
|first=Helmut
|last=Hasse
|authorlink=Helmut Hasse
|title=Vorlesungen über Zahlentheorie
|edition=2nd revised
|series=Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen
|volume=59
|publisher=[[Springer-Verlag]]
|year=1964
|mr=0188128 | zbl=0123.04201
}} see chapter 13.
* {{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 }}
* {{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 }}
* {{Cite journal
|first1=Robert
|last1=Spira
|title=Calculation of Dirichlet L-Functions
|journal=Mathematics of Computation
|volume=23
|pages=489–497
|year=1969
|doi=10.1090/S0025-5718-1969-0247742-X
|mr=0247742 | zbl=0182.07001
|issue=107
}}
* {{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 }}

==External links==
* {{springer|title=Dirichlet character|id=p/d032810}}

[[Category:Zeta and L-functions]]

[[de:Charakter (Mathematik)#Dirichlet-Charaktere]]

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