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>Template:Cite journal</ref> 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>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.<ref>Template:Harvnb</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>Template:Harvnb</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>Template:Harvnb</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>Template:Harvnb</ref><ref>Template:Harvnb</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>Template:Harvnb</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>Template:Harvnb</ref><ref>Template:Harvnb</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">Template:Harvnb</ref><ref name="IwaniecKowalski84">Template:Harvnb</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 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">Template:Harvnb</ref><ref name="IwaniecKowalski84" />
For generalizations, see the article on functional equations of L-functions.
Zeros
Let <math> \chi </math> be a primitive character modulo <math> q </math>, with <math> q>1 </math>.
There are no 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 integers <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">Template:Harvnb</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>Template:Cite book</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>Template:Harvnb</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
- Special values of L-functions