Editing Dirichlet L-function

{{Short description|Type of mathematical function}}
{{DISPLAYTITLE:Dirichlet ''L''-function}}
In [[mathematics]], a '''Dirichlet''' <math>L</math>-'''series''' is a function of the form

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

where <math> \chi </math> is a [[Dirichlet character]] and <math> s </math> a [[complex variable]] with [[real part]] greater than <math> 1 </math>. It is a special case of a [[Dirichlet series]]. By [[analytic continuation]], it can be extended to a [[meromorphic function]] on the whole [[complex plane]], and is then called a '''Dirichlet <math> L </math>-function''' and also denoted <math> L ( s , \chi) </math>.

These functions are named after [[Peter Gustav Lejeune Dirichlet]] who introduced them in {{harv|Dirichlet|1837}} to prove the [[Dirichlet's theorem on arithmetic progressions|theorem on primes in arithmetic progressions]] that also bears his name. In the course of the proof, Dirichlet shows that <math> L ( s , \chi) </math> is non-zero at <math> s = 1 </math>. Moreover, if <math> \chi </math> is principal, then the corresponding Dirichlet <math> L </math>-function has a [[simple pole]] at <math> s = 1 </math>. Otherwise, the <math> L </math>-function is [[entire function|entire]].

==Euler product==
Since a Dirichlet character <math> \chi </math> is [[completely multiplicative]], its <math> L </math>-function can also be written as an [[Euler product]] in the [[half-plane]] of [[absolute convergence]]:
:<math>L(s,\chi)=\prod_p\left(1-\chi(p)p^{-s}\right)^{-1}\text{ for }\text{Re}(s) > 1,</math>
where the product is over all [[prime number]]s.<ref>{{harvnb|Apostol|1976|loc=Theorem 11.7}}</ref>

==Primitive characters==

Results about ''L''-functions are often stated more simply if the character is assumed to be primitive, although the results typically can be extended to imprimitive characters with minor complications.<ref>{{harvnb|Davenport|2000|loc=chapter 5}}</ref> This is because of the relationship between a imprimitive character <math>\chi</math> and the primitive character <math>\chi^\star</math> which induces it:<ref>{{harvnb|Davenport|2000|loc=chapter 5, equation (2)}}</ref>
:<math>
  \chi(n) =
    \begin{cases}
      \chi^\star(n), & \mathrm{if} \gcd(n,q) = 1 \\
      0, & \mathrm{if} \gcd(n,q) \ne 1
    \end{cases}
</math>
(Here, ''q'' is the modulus of ''χ''.) An application of the Euler product gives a simple relationship between the corresponding ''L''-functions:<ref>{{harvnb|Davenport|2000|loc=chapter 5, equation (3)}}</ref><ref>{{harvnb|Montgomery|Vaughan|2006|p=282}}</ref>
:<math>
  L(s,\chi) = L(s,\chi^\star) \prod_{p \,|\, q}\left(1 - \frac{\chi^\star(p)}{p^s} \right)
</math>
(This formula holds for all ''s'', by analytic continuation, even though the Euler product is only valid when Re(''s'') > 1.) The formula shows that the ''L''-function of ''χ'' is equal to the ''L''-function of the primitive character which induces ''χ'', multiplied by only a finite number of factors.<ref>{{harvnb|Apostol|1976|p=262}}</ref>

As a special case, the ''L''-function of the principal character <math>\chi_0</math> modulo ''q'' can be expressed in terms of the [[Riemann zeta function]]:<ref>{{harvnb|Ireland|Rosen|1990|loc=chapter 16, section 4}}</ref><ref>{{harvnb|Montgomery|Vaughan|2006|p=121}}</ref>
:<math>
  L(s,\chi_0) = \zeta(s) \prod_{p \,|\, q}(1 - p^{-s})
</math>

==Functional equation==

Dirichlet ''L''-functions satisfy a [[functional equation]], which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of <math>L(s,\chi)</math> to the value of <math>L(1-s, \overline{\chi})</math>. Let ''χ'' be a primitive character modulo ''q'', where ''q'' > 1. One way to express the functional equation is:<ref name="MontgomeryVaughan333" />
:<math>L(s,\chi) = W(\chi) 2^s \pi^{s-1} q^{1/2-s} \sin \left( \frac{\pi}{2} (s + \delta) \right)  \Gamma(1-s) L(1-s, \overline{\chi}).</math>
In this equation, Γ denotes the [[gamma function]]; 
:<math>\chi(-1)=(-1)^{\delta}</math> ; and
:<math>W(\chi) = \frac{\tau(\chi)}{i^{\delta} \sqrt{q}}</math>
where ''τ''{{hairsp}}({{hairsp}}''χ'') is a [[Gauss sum]]:
:<math>\tau(\chi) = \sum_{a=1}^q \chi(a)\exp(2\pi ia/q).</math>
It is a property of Gauss sums that |''τ''{{hairsp}}({{hairsp}}''χ''){{hairsp}}| = ''q''<sup>1/2</sup>, so |''W''{{hairsp}}({{hairsp}}''χ''){{hairsp}}| = 1.<ref name="MontgomeryVaughan332">{{harvnb|Montgomery|Vaughan|2006|p=332}}</ref><ref name="IwaniecKowalski84">{{harvnb|Iwaniec|Kowalski|2004|p=84}}</ref>

Another way to state the functional equation is in terms of
:<math>\Lambda(s,\chi) = q ^{s/2} \pi^{-(s+\delta)/2} \operatorname{\Gamma}\left(\frac{s+\delta}{2}\right) L(s,\chi).</math>
The functional equation can be expressed as:<ref name="MontgomeryVaughan333" /><ref name="IwaniecKowalski84" />
:<math>\Lambda(s,\chi) = W(\chi) \Lambda(1-s,\overline{\chi}).</math>

The functional equation implies that <math>L(s,\chi)</math> (and <math>\Lambda(s,\chi)</math>) are [[entire function|entire functions]] of ''s''. (Again, this assumes that ''χ'' is primitive character modulo ''q'' with ''q'' > 1. If ''q'' = 1, then <math>L(s,\chi) = \zeta(s)</math> has a pole at ''s'' = 1.)<ref name="MontgomeryVaughan333">{{harvnb|Montgomery|Vaughan|2006|p=333}}</ref><ref name="IwaniecKowalski84" />

For generalizations, see: [[Functional equation (L-function)]].

==Zeros==
[[Image:Mplwp dirichlet beta.svg|thumb|right|300px|The Dirichlet ''L''-function ''L''(''s'', ''χ'') = 1 − 3<sup>−''s''</sup> + 5<sup>−''s''</sup> − 7<sup>−''s''</sup> + ⋅⋅⋅ (sometimes given the special name [[Dirichlet beta function]]), with trivial zeros at the negative odd integers]]

Let ''χ'' be a primitive character modulo ''q'', with ''q'' > 1.

There are no [[zero of a function|zeros]] of ''L''(''s'', ''χ'') with Re(''s'') > 1. For Re(''s'') < 0, there are zeros at certain negative [[integer]]s ''s'':
* If ''χ''(−1) = 1, the only zeros of ''L''(''s'', ''χ'') with Re(''s'') < 0 are simple zeros at −2, −4, −6, .... (There is also a zero at ''s'' = 0.) These correspond to the poles of <math>\textstyle \Gamma(\frac{s}{2})</math>.<ref name="DavenportCh9">{{harvnb|Davenport|2000|loc=chapter 9}}</ref>
* If ''χ''(−1) = −1, then the only zeros of ''L''(''s'', ''χ'') with Re(''s'') < 0 are simple zeros at −1, −3, −5, .... These correspond to the poles of <math>\textstyle \Gamma(\frac{s+1}{2})</math>.<ref name="DavenportCh9" />
These are called the trivial zeros.<ref name="MontgomeryVaughan333"/>

The remaining zeros lie in the critical strip 0 ≤ Re(''s'') ≤ 1, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line Re(''s'') = 1/2. That is, if <math>L(\rho,\chi)=0</math> then <math>L(1-\overline{\rho},\chi)=0</math> too, because of the functional equation. If ''χ'' is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if ''χ'' is a complex character. The [[generalized Riemann hypothesis]] is the conjecture that all the non-trivial zeros lie on the critical line Re(''s'') = 1/2.<ref name="MontgomeryVaughan333" />

Up to the possible existence of a [[Siegel zero]], zero-free regions including and beyond the line Re(''s'') = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet ''L''-functions: for example, for ''χ'' a non-real character of modulus ''q'', we have

:<math> \beta < 1 - \frac{c}{\log\!\!\; \big(q(2+|\gamma|)\big)} \ </math>

for β + iγ a non-real zero.<ref>{{cite book |last=Montgomery |first=Hugh L. |author-link=Hugh Montgomery (mathematician) |title=Ten lectures on the interface between analytic number theory and harmonic analysis |series=Regional Conference Series in Mathematics |volume=84 |location=Providence, RI |publisher=[[American Mathematical Society]] |year=1994 |isbn=0-8218-0737-4 |zbl=0814.11001 |page=163}}</ref>

== Relation to the Hurwitz zeta function ==
The Dirichlet ''L''-functions may be written as a linear combination of the [[Hurwitz zeta function]] at rational values. Fixing an integer ''k'' ≥ 1, the Dirichlet ''L''-functions for characters modulo ''k'' are linear combinations, with constant coefficients, of the ''ζ''(''s'',''a'') where ''a'' = ''r''/''k'' and ''r'' = 1, 2, ..., ''k''. This means that the Hurwitz zeta function for rational ''a'' has analytic properties that are closely related to the Dirichlet ''L''-functions. Specifically, let ''&chi;'' be a character modulo ''k''.  Then we can write its Dirichlet ''L''-function as:<ref>{{harvnb|Apostol|1976|p=249}}</ref>

:<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}
= \frac{1}{k^s} \sum_{r=1}^k \chi(r) \operatorname{\zeta}\left(s,\frac{r}{k}\right).</math>

==See also==

*[[Generalized Riemann hypothesis]]
*[[L-function]]
*[[Modularity theorem]]
*[[Artin conjecture (L-functions)|Artin conjecture]]
*[[Special values of L-functions]]

==Notes==
{{reflist}}

== References ==
* {{Apostol IANT}}
*{{dlmf|id=25.15|first=T. M.|last=Apostol}}
* {{cite book|first=H.|last=Davenport|author-link=Harold Davenport
|title=Multiplicative Number Theory
|publisher=Springer
|year=2000
|edition=3rd
|isbn=0-387-95097-4}}
* {{Cite journal
| last=Dirichlet
| first=P. G. L.
| author-link=Peter Gustav Lejeune Dirichlet
| title=Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält
| journal=Abhand. Ak. Wiss. Berlin
| volume=48
| year=1837
}}
* {{cite book|first1=Kenneth|last1=Ireland|first2=Michael|last2=Rosen|author-link2=Michael Rosen (mathematician)|title=A Classical Introduction to Modern Number Theory|edition=2nd|publisher=Springer-Verlag|year=1990}}
* {{cite book|first1=Hugh L.|last1=Montgomery |author-link=Hugh Montgomery (mathematician)|first2=Robert C.|last2=Vaughan |author-link2=Robert Charles Vaughan (mathematician) | title=Multiplicative number theory. I. Classical theory| series=Cambridge tracts in advanced mathematics| volume=97| publisher=Cambridge University Press|year=2006| isbn=978-0-521-84903-6}}
* {{cite book
|last1=Iwaniec
|first1=Henryk
|author-link=Henryk Iwaniec
|last2=Kowalski
|first2=Emmanuel
|year=2004
|title=Analytic Number Theory
|series=American Mathematical Society Colloquium Publications
|volume=53
|location=Providence, RI
|publisher=American Mathematical Society
}}
* {{springer|title=Dirichlet-L-function|id=p/d032890}}

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[[Category:Zeta and L-functions]]