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{{Short description|Type of mathematical function}}
{{DISPLAYTITLE:Dirichlet ''L''-function}}
In [[mathematics]], a '''Dirichlet ''L''-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]]; it is then called a '''Dirichlet ''L''-function'''.

These functions are named after [[Peter Gustav Lejeune Dirichlet]] who introduced them in 1837<ref>{{Cite journal |last=Dirichlet |first=Peter Gustav Lejeune |author-link=Peter Gustav Lejeune Dirichlet |date=1837 |title=Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält |url=https://doi.org/10.1017/CBO9781139237321.012 |journal=Abhand. Ak. Wiss. Berlin |volume=48}}</ref> to prove his [[Dirichlet's theorem on arithmetic progressions|theorem on primes in arithmetic progressions]]. In his proof, Dirichlet showed 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 ''L''-function has a [[simple pole]] at <math> s = 1 </math>. Otherwise, the ''L''-function is [[entire function|entire]].

==Euler product==
Since a Dirichlet character <math> \chi </math> is [[completely multiplicative]], its ''L''-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{otherwise}.
\end{cases}
</math>
(Here, <math> q </math> is the modulus of <math> \chi </math>.) 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>
By analytic continuation, this formula holds for all complex <math>
s
</math>, even though the Euler product is only valid when <math>
\operatorname{Re}(s)>1
</math>. The formula shows that the ''L''-function of <math> \chi </math> is equal to the ''L''-function of the primitive character which induces <math> \chi </math>, 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 <math> q </math> 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 values of <math>L(s,\chi)</math> to the values of <math>L(1-s, \overline{\chi})</math>.

Let ''<math> \chi </math>'' be a primitive character modulo <math> q </math>, where <math>
q>1
</math>. One way to express the functional equation is as<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>
where <math>
\Gamma
</math> is the [[gamma function]], <math>
\chi(-1)=(-1)^{\delta}
</math>, and
:<math>W(\chi) = \frac{\tau(\chi)}{i^{\delta}\sqrt{q}},</math>
where <math>\tau(\chi)</math> is the [[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 <math>|\tau(\chi)| = \sqrt{q} </math>, so <math>|W(\chi)| = 1 </math>.<ref name="MontgomeryVaughan332">{{harvnb|Montgomery|Vaughan|2006|p=332}}</ref><ref name="IwaniecKowalski84">{{harvnb|Iwaniec|Kowalski|2004|p=84}}</ref> Another functional equation is
:<math>\Lambda(s,\chi) = q ^{s/2} \pi^{-(s+\delta)/2} \operatorname{\Gamma}\left(\frac{s+\delta}{2}\right) L(s,\chi),</math>
which can be expressed as<ref name="MontgomeryVaughan333" /><ref name="IwaniecKowalski84" />
:<math>\Lambda(s,\chi) = W(\chi) \Lambda(1-s,\overline{\chi}).</math>

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

For generalizations, see the article on [[Functional equation (L-function)|functional equations of ''L''-functions]].

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

Let <math> \chi </math> be a primitive character modulo <math> q </math>, with <math> q>1 </math>.

There are no [[zero of a function|zeros]] of <math>L(s,\chi)</math> with <math>
\operatorname{Re}(s)>1
</math>. For <math>
\operatorname{Re}(s) < 0
</math>, there are zeros at certain negative [[integer]]s <math>s</math>:
* If <math> \chi(-1) = 1 </math>, the only zeros of <math>L(s,\chi)</math> with <math>
\operatorname{Re}(s) < 0
</math> are simple zeros at <math>-2,-4,-6,\dots</math> There is also a zero at <math>s = 0</math> when <math> \chi </math> is non-principal. These correspond to the poles of <math>\textstyle \Gamma(\frac{s}{2})</math>.<ref name="DavenportCh9">{{harvnb|Davenport|2000|loc=chapter 9}}</ref>
* If <math> \chi(-1) = -1 </math>, then the only zeros of <math>L(s,\chi)</math> with <math>
\operatorname{Re}(s) < 0
</math> are simple zeros at <math>-1,-3,-5,\dots</math> 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 <math>
0 \leq \operatorname{Re}(s) \leq 1
</math>, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line <math>
\operatorname{Re}(s) = 1/2
</math>. 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 <math> \chi </math> is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if <math> \chi </math> is a complex character. The [[generalized Riemann hypothesis]] is the conjecture that all the non-trivial zeros lie on the critical line <math>
\operatorname{Re}(s) = 1/2
</math>.<ref name="MontgomeryVaughan333" />

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

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

for <math> \beta + i\gamma </math> 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 ==
Dirichlet ''L''-functions may be written as linear combinations of the [[Hurwitz zeta function]] at rational values. Fixing an integer <math>
k \geq 1
</math>, Dirichlet ''L''-functions for characters modulo <math> k </math> are linear combinations with constant coefficients of the <math> \zeta(s,a) </math> where <math> a = r/k </math> and ''<math> r = 1,2,\dots,k </math>''. This means that the Hurwitz zeta function for rational <math> a </math> has analytic properties that are closely related to the Dirichlet ''L''-functions. Specifically, if <math> \chi </math> is a character modulo <math> k </math>, 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]]

Dirichlet L-function - Revision history

imported>JJMC89 bot III: Removing :Category:Eponymous functions per Wikipedia:Categories for discussion/Log/2025 October 27#Eponyms in mathematics round 2