In mathematics, a Dirichlet L-series is a function of the form
<math display="block">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 to prove his 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.
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 display="block">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 numbers.
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. This is because of the relationship between a imprimitive character <math>\chi</math> and the primitive character <math>\chi^\star</math> which induces it:
<math display="block">
\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:
<math display="block">
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.
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:
<math display="block">
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 Another functional equation is
<math display="block">\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
- 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>.
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
<math display="block">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
- Special values of L-functions